Unit 4 Chapter 4 Trigonometry and the Unit Circle 4.1 ... · PDF fileUnit 4 Math 3200 1 x y Ter m na Arm a I nt a Ar m Unit 4 Chapter 4 Trigonometry and the Unit Circle 4.1 Angles

Unit 4 Math 3200 1

x

y

Ter m inal Ar m

a I nit ial Ar m

Unit 4 Chapter 4

Trigonometry and the Unit Circle

41 Angles and Angle Measure

(I) Angles in Standard Position

Reference Angle

formed between the x-axis and

the terminal arm

Terminal Arm

can rotate clockwise (ndash rotation)

or counterclockwise (+ rotation)

Angle of Rotation

Goals

Angles in Standard Position

Angles Measured in Degrees and Radians

Converting between Degree and Radian Measure

Coterminal Angles

Solving Problems involving Arc Length

Unit 4 Math 3200 2

Example Sketch each angle in standard position

1 150deg 2 270deg 3 510deg

4 ndash210deg 5 ndash720deg

(II) Angles measured in Degrees and Radians

Degree Measure

Radian Measure

What is radian measure

One revolution is equivalent

to the circumference C = 2πr

If we consider the unit circle


-have center at the origin and

initial arm along the positive

x-axis

1 revolution = 360deg

Unit 4 Math 3200 3

where r = 1 then one revolution is equivalent to

Example Sketch each radian measure of rotation

1 2π radians 2 1π or π radians 3 1

2120587 119900119903

120587

2 119903119886119889119894119886119899119904

C = 2π(1) radians

= 2π radians

Therefore

2π radians = 1 revolution

Comparing Radian Measure to Degree Measure

1 revolution 2π radians = _______

1

2 revolution π radians = _______

Unit 4 Math 3200 4

(III) Converting Between Degree and Radian Measure

Example Change each degree measure to radians each radian

measure to degrees and sketch angle in standard

position

(a) 60deg (b) minus3120587

4

(c) 7120587

6 (d) ndash45

Unit 4 Math 3200 5

(IV) Determining the measure of angles that are

coterminal with a given angle

Sketch each angle below in standard position on the same grid

(a) 30deg (b) 750deg (c) ndash330deg

Example State one positive and one negative angle coterminal

with

(a) 210deg (b) minus7120587

5

Coterminal Angles

Angles in standard position that have the same _____________

Angles that are a multiple of ___________ away

Unit 4 Math 3200 6

Coterminal Angles in General Form

Example

Express each of the given angles in general form and state the angles

that are coterminal with the given angle that also satisfy the stated

domain

(a) 135deg domain ndash720deg le Ө le 720deg

(b) 5120587

6 domain ndash4π le Ө le 4π

Unit 4 Math 3200 7

(V) Solving Problems involving arc lengths central

angles and the radius in a circle

Arc Length of a Circle

All arcs that subtend the same central angle

have different arc lengths depending on the

radius of the circle The arc length is proportional

to the radius This is true for any central angle

and related arc length

Consider the circle with radius r and the sector with

central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903

Arc length of a circle

a =

Unit 4 Math 3200 8

Example

Determine arc length in a circle

Rosemarie is taking a course in industrial engineering

For an assignment she is designing the interface of

a DVD player In her plan she includes a decorative

arc below the onoff button The arc has central angle

130deg in a circle with radius 67 mm Determine the

length of the arc to the nearest tenth of a mm

Textbook Homework pg 175 1-9 12-14 16-17 21-23 C4

Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle

Has center at the origin (0 0) and a radius length r = 1

Label the coordinates of the four terminal positions on the unit

circle below

x

y

Goals

The Unit Circle

Developing and Applying the Equation of the Unit Circle

Locating Coordinates of Points on the Unit Circle

Unit 4 Math 3200 10

(II) Developing and Applying the Equation of the

Unit Circle

Equation of a Circle Centered at the Origin

(a) Identify the radius of the circle

(b) Identify the coordinates of the point P

(c) Draw a vertical line segment PN from P to the x-axis

(d) Draw a horizontal line segment NO from N to the origin

(e) What type of triangle have you created

Unit 4 Math 3200 11

(f) Determine the lengths of PN and NO

(g) Write an equation that relates the three sides of the triangle

(h) Write an equation of any circle that is centered at the origin

and has a radius of lsquorrsquo

Example

Determine the equation of the circle with center at the origin and

(a) r = radic10 (b) diameter = 12 (c) passing through the

point (8 15)


Any circle with its center at the origin O(0 0)

and radius lsquorrsquo is defined by the equation

________________

Unit 4 Math 3200 12

(III) Using Symmetry and Patterns to Locate the

Coordinates of Points on the Unit Circle

Example

Determine Coordinates for Points of the Unit Circle

(a) 119875 (1

5 119910)in quadrant 1

Remember The Unit Circle

Coordinates on the unit circle satisfy the

equation x2 + y2 = 1

x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14

Relating Arc Length and Angle Measure in Radians

lsquoarsquo is the arc length

Ө is the central angle in radians

r is the radius (which is measured in the same units as lsquoarsquo)

In the unit circle (r = 1) the formula a = Өr

becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө

This means that the central angle

and the subtended arc on the unit

circle have the same numerical

value

Unit 4 Math 3200 15

Example Multiples of 120645

120785 on the Unit Circle

(a) On a diagram of the unit circle show the integral multiples

of π

3 in the interval 0 le Ө le 2π

(b) What are the coordinates for each point P(Ө) in part (a)

(c) Identify any patterns observed in the coordinates of the point

x

y

Unit 4 Math 3200 16

Note

You can see patterns to determine coordinates of points The

numerical value of the coordinates of points change sign

Some points on the unit circle correspond to exact values of special

angles (where reference angles equal 30deg 45deg and 60deg)

If P(a b) is in quadrant I then both lsquoarsquo and lsquobrsquo are positive P(Ө + π)

is in quadrant III Its coordinates are (ndasha ndashb) where a gt 0 and b gt 0

Unit 4 Math 3200 17

Example Determine the exact coordinates of the terminal arm

of angle Ө given P(Ө)

(a) 119875 (5120587

6) (b) 119875 (

3120587

2)

Example Identify the measure for the central angle Ө given in

the interval 0 le Ө lt 2π such that P(Ө) is the given

point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y

Textbook Homework pg 186 1-6 10-13 16

Unit 4 Math 3200 18

43 Trigonometric Ratios (Part I)

(I) Primary Trig Ratios and Coordinates of Points on Terminal Arm

If P(Ө) = (x y) is a point on the terminal arm of Ө then define the

coordinates of the point P(Ө) in terms of the primary ratios based

on the diagram below

Remember Primary Trig Ratios

SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =

From above we can see that

x = and y =

hence the point P(Ө) = (x y)

Can be expressed as P(Ө) =

Goals

Primary Trig Ratios amp Coordinates of Points on Terminal Arm

Reciprocal Trig Ratios

Determining Trig Ratios of Point on Terminal Arm amp

Determining Angle of Rotation

Exact Values of Trig Ratios

Approximate Values of Trig Ratios

Unit 4 Math 3200 19

From this we can also conclude that since tan 120579 = 119910

119909

then

(II) Reciprocal Trig Ratios

Coordinates of Point on Terminal Arm

Since x = and y =

Any point on the terminal arm of angle Ө

can be represented by P(Ө) = (cos Ө sin Ө)

tan Ө =

Each primary trig ratio has a reciprocal trig ratio

(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20

(III) Determining Trig Ratios Given a Point on the Terminal Arm

and Determining Angle of Rotation

Example Given that each point below is a point that lies on the

unit circle and the terminal arm of Ө

(i) Sketch a diagram

(ii) Determine all trig ratios

(iii) Determine the angle of rotation from standard position

(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21

(IV) Exact Values of Trig Ratios

Exact values for the trigonometric ratios can be determined using

special triangles (30deg-60deg-90deg) or (45deg-45deg-90deg) and multiples of

120579 = 0120587

6

120587

4

120587

3

120587

2 or θ = 0deg 30deg 45deg 60deg 90deg for points P(θ) on

the unit circle

Special angles of rotation on the unit circle occur in terminal positions

where the reference angle is 30deg 45deg and 60deg

Unit 4 Math 3200 22

If we consider the first quadrant and the terminal positions for

30deg 45deg and 60deg we can develop the side lengths in the resulting

triangle

Unit 4 Math 3200 23

For the unit circle identify the EXACT COORDINATES of each point

indicated

Exact Values of Special Angles

(A) 30deg ndash 60deg ndash 90deg Triangles

(B) 45deg ndash 45deg ndash 90deg Triangles

Note the pattern

Unit 4 Math 3200 24

Example Determining an EXACT VALUE given a trig ratio

Include a sketch to determine the EXACT VALUE for each trig ratio

(a) sin (minus4120587

3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25

Example Determine the EXACT VALUE of each trig expression

(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26

(V) Approximate Values of Trig Ratios

Approximate values for sine cosine and tangent can be

determined for angles measured in degrees or radians The mode of

the calculator can be set to degrees or radians

cos 60deg = _____ (degree mode) sin (-30deg) = _____ (degree mode)

cos 60 = _____ (radian mode) sin (-30) = _____ (radian mode)

To evaluate cosecant secant or cotangent using the correct reciprocal

relationship For example to determine the value of sec 33 you would

use the ldquocosrdquo function on your calculator Thus

13 3 sec

Example Determine the approximate value (4 decimal places) for

(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27

43 Trigonometric Ratios (Part II)

Determining Where a Trig Ratio is Positive or Negative

The placement of a terminal arm in a quadrant will implicate that

some trig ratios will be positive and others negative

x

y

Goals


Determining Angles of a Trig Ratio over a Specified Domain

Given a Trig Ratio Determine other Trig Ratios

Given a Point on Terminal Arm Determine over a Specified Domain

Given Points on Terminal Arm Determine Arc Length

Unit 4 Math 3200 28

Example Identify the quadrant(s) where

(a) sin Ө gt 0 (b) tan Ө lt 0 (c) csc Ө lt 0

(d) sec Ө gt 0 and cot Ө lt 0

(VI) Determining angles given a trig ratio over a specified domain

Example Determine all of the angles within the specified domain

that satisfies the given trig ratio

(a) sec 120579 = minusradic2 where ndash2π le Ө le 2π

x

y

Unit 4 Math 3200 29

(b) csc 120579 = minus2 where ndash180deg le Ө lt 180deg

(c) tan Ө = ndash24567 where 0 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 30

(VII) Given a trig ratio determine other trig ratios

Example Determine the EXACT VALUE of the other five trig

ratios under the given conditions

(a) sin 120579 = minus1

2 where 180deg le Ө lt 270deg

(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31

(VIII) Given a point on the terminal arm determine Ө in the

specified domain

Example P(ndash5 12) where ndash360deg le Ө lt 360deg

(IX) Given points on the terminal arm determine arc length

Example Determine the length of 119860 to the nearest

tenth of a unit

x

y

B (-86)A (-68)

Q

P201 ndash P202 3 ndash 7 10 ndash 12 18 19 21 C2

Unit 4 Math 3200 32

44 Introduction to Trigonometric Equations

Remember ndash Placement of a Terminal Arm based on a Trig Ratio

Example Where is

(a) csc Ө lt 0 (b) tan Ө gt 0

(I) Stating a Domain using Set Notation vs Interval Notation

When we use set notation to express a domain such as 0 le Ө le π

It can also be expressed using interval notation as

Therefore Ө є (0 π] is the same as _____________

Express ndashπ lt Ө le 2π using interval notation ____________

x

y

A

C

S

T

C ndash Cosine (amp Sec) are positive

A ndash All trig ratios are positive

S ndash Sine (amp Csc) are positive

T ndash Tangent (amp Cot) are positive

Goals

Stating a Domain Using Set Notation vs Interval Notation

Solving Linear Trig Equations

Solving Quadratic Trig Equations

Unit 4 Math 3200 33

(II) Solving Linear Trig Equations a + b = 0

So far we have solved trig ratios where we havenrsquot had to isolate

the trig ratio

Example Solve for Ө where csc Ө = ndash radic2 where Ө є [0 2π]

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34

Sometimes we have to isolate the trig ratio to solve

Example Solve for all possible values of Ө where

radic3sec Ө + 2= 0


5sin Ө + 2 = 1 + 3sin Ө 0deg le Ө lt 360deg

x

y

x

y

Unit 4 Math 3200 35

(III) Solving Quadratic Trig Equations a + b +c=0

Example Solve the following trig equations for the specified

domain

(a) 2 sin2Ө = ndash 5 sin Ө + 3 Ө є (ndashπ 2π)

Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36

(b) 5 sec2Ө = 12 + 4sec Ө for all Ө in radians

Give solutions as exact values where possible Otherwise

give approximate measures to the nearest thousandth of a

degree

x

y

x

y

Unit 4 Math 3200 37

(c) 2 csc2Ө ndash 8 = 0 for all Ө in radians



degree

x

y

x

y

Unit 4 Math 3200 38

(d) Given g(Ө) = cos2Ө ndash 3 and p(Ө) = 2 cos Ө determine the

values of Ө such that g(Ө) = p(Ө) where Ө є [0 4π]



degree

x

y

x

y

Textbook Homework pg 211 1-5 9-11 16 17 1822

Unit 4 Math 3200 2

Example Sketch each angle in standard position

1 150deg 2 270deg 3 510deg

4 ndash210deg 5 ndash720deg

(II) Angles measured in Degrees and Radians

Degree Measure

Radian Measure

What is radian measure

One revolution is equivalent

to the circumference C = 2πr

If we consider the unit circle


-have center at the origin and

initial arm along the positive

x-axis

1 revolution = 360deg

Unit 4 Math 3200 3




2120587 119900119903

120587

2 119903119886119889119894119886119899119904

C = 2π(1) radians

= 2π radians

Therefore




1


Unit 4 Math 3200 4




position

(a) 60deg (b) minus3120587

4

(c) 7120587

6 (d) ndash45

Unit 4 Math 3200 5






with

(a) 210deg (b) minus7120587

5

Coterminal Angles



Unit 4 Math 3200 6


Example



domain


(b) 5120587


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 3




2120587 119900119903

120587

2 119903119886119889119894119886119899119904

C = 2π(1) radians

= 2π radians

Therefore




1


Unit 4 Math 3200 4




position

(a) 60deg (b) minus3120587

4

(c) 7120587

6 (d) ndash45

Unit 4 Math 3200 5






with

(a) 210deg (b) minus7120587

5

Coterminal Angles



Unit 4 Math 3200 6


Example



domain


(b) 5120587


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 4




position

(a) 60deg (b) minus3120587

4

(c) 7120587

6 (d) ndash45

Unit 4 Math 3200 5






with

(a) 210deg (b) minus7120587

5

Coterminal Angles



Unit 4 Math 3200 6


Example



domain


(b) 5120587


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 5






with

(a) 210deg (b) minus7120587

5

Coterminal Angles



Unit 4 Math 3200 6


Example



domain


(b) 5120587


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 6


Example



domain


(b) 5120587


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 7










central angle Ө

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

d

E

F

120579 = 119886119903119888 119897119890119899119892119905ℎ

119903119886119889119894119906119904

O

C

B

a θ

r

120579 = 119886

119903


a =

Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 8

Example









Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 9

42 The Unit Circle

(I) The Unit Circle

The Unit Circle



circle below

x

y

Goals

The Unit Circle



Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 10


Unit Circle







Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 11





Example



point (8 15)




________________

Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 12



Example


(a) 119875 (1





x

y

Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 13

Example


(b) 119875 (119909 minus3

8)

x

y

Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 14






becomes a = Ө1

a = Ө


a = Өr O

C

B

a θ

r

O

C

B

Ө

1

Ө




value

Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 15




of π




x

y

Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 16

Note







Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 17



(a) 119875 (5120587

6) (b) 119875 (

3120587

2)



point

(a) (radic2

2 minus

radic2

2) (b) (

radic3

2 minus

1

2)

x

y

x

y

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 18







SOHCAHTOA

sin 120579 = 119900119901119901

ℎ119910119901 sin 120579 =

cos 120579 = 119886119889119895

ℎ119910119901 cos 120579 =


x = and y =



Goals







Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 19


119909

then



Since x = and y =



tan Ө =


(i) sin 120579 = 119900119901119901

ℎ119910119901 cosecant Ө csc 120579 =

ℎ119910119901

119900119901119901=

1

119910=

1

sin 120579

(ii) cos 120579 = 119886119889119895

ℎ119910119901 secant Ө sec 120579 =

ℎ119910119901

119886119889119895=

1

119909=

1

cos 120579

(iii) tan 120579 = 119900119901119901

119886119889119895 cotangant Ө cot 120579 =

119886119889119895

119900119901119901=

119909

119910=

cos 120579

sin 120579

Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 20








(a) 119875 (5

13 minus

12

13) (b) 119875 (minus

1

2

radic3

2)

x

y

x

y

Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 21




120579 = 0120587

6

120587

4

120587

3

120587


the unit circle



Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 22



triangle

Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 23


indicated




Note the pattern

Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 24




3) (b) sec (

7120587

6)

(c) cot 270deg

x

y

x

y

x

y

Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 25


(a) csc2 (120587

3) + cot2 (

11120587

4) (b)

cot(120587

3)+cos(

11120587

3)

csc(minus240deg)

Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 26










13 3 sec


(a) csc (7120587

5) (b) cot (23) (c) cot (210deg)

P201 ndash P202 1 2 8 9

Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 27





x

y

Goals






Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 28








x

y

Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 29



x

y

x

y

Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 30




(a) sin 120579 = minus1


(b) sec Ө = 55 where 3120587

2 le Ө le 2π

x

y

x

y

Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 31


specified domain




tenth of a unit

x

y

B (-86)A (-68)

Q


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 32



Example Where is







x

y

A

C

S

T





Goals




Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 33



the trig ratio


sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 34



radic3sec Ө + 2= 0



x

y

x

y

Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 35



domain


Sin2 Ө

cos2 Ө

tan2 Ө

csc2 Ө

sec2 Ө

cot2 Ө

sin Ө

cos Ө

tan Ө

csc Ө

sec Ө

cot Ө

x

y

x

y

Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 36




degree

x

y

x

y

Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 37




degree

x

y

x

y

Unit 4 Math 3200 38





degree

x

y

x

y


Unit 4 Math 3200 38





degree

x

y

x

y


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Unit 4 Chapter 4 Trigonometry and the Unit Circle 4.1 ... · PDF fileUnit 4 Math 3200 1 x y Ter m na Arm a I nt a Ar m Unit 4 Chapter 4 Trigonometry and the Unit Circle 4.1 Angles