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Trigonometric Functions of Any Angle&

Polar Coordinates

Sections 8.1, 8.2, 8.3, 21.10

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Definitions of Trigonometric Functions of Any Angle

Let be an angle in standard position with (x, y) a point on the terminal side of and

Definitions of Trig Functions of Any Angle(Sect 8.1)

2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

y

x

(x, y)

r

3

Since the radius is always positive (r > 0), the signs

of the trig functions are dependent upon the signs of

x and y. Therefore, we can determine the sign of

the functions by knowing the quadrant in which the

terminal side of the angle lies.

The Signs of the Trig Functions

4

The Signs of the Trig Functions

5

To determine where each trig function is POSITIVE:

A

CT

S

“All Students Take Calculus”

Translation:

A = All 3 functions are positive in Quad 1

S= Sine function is positive in Quad 2

T= Tangent function is positive in Quad 3

C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, and cotangent is positive wherever tangent is positive.

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Determine if the following functions are positive or negative:

Example

1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

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Examples

For the given values, determine the quadrant(s) in which the terminal side of θ lies.

1) sin 0.3614 2) tan 2.553 3) cos 0.866

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Examples

Determine the quadrant in which the terminal side of θ lies, subject to both given conditions.

1) sin 0, cos 0 2) sec 0, cot 0

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ExamplesFind the exact value of the six trigonometric functions of θ if

the terminal side of θ passes through point (3, -5).

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The values of the trig functions for non-acute angles can be found using the values of the corresponding reference angles.

Reference Angles (Sect 8.2)

Definition of Reference Angle

Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis.

ref

Quadrant II

ref

Quadrant IV

ref

Quadrant III

ref

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Example

Find the reference angle for 225

Solution y

x

ref

By sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.

ref

ref

ref

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So what’s so great about reference angles?

Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.

For example,

1sin 225 (sin 45 )

2

45° is the ref angleIn Quad 3, sin is negative

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Example

Give the exact value of the trig function (without using a calculator).

cos 150

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Examples (Text p 239 #6)

Express the given trigonometric function in terms of the same function of a positive acute angle.

6 ) tan 91 )sec 345a b

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Examples (Text p 239 #8)

Express the given trigonometric function in terms of the same function of a positive acute angle.

8 ) cos 190 )cot 290a b

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Now, of course you can simply use the calculator to find the value of the trig function of any angle and it will correctly return the (approximate) answer with the correct sign.

Remember:

Make sure the Mode setting is set to the correct form of the angle: Radian or Degree

To find the trig functions of the reciprocal functions (csc, sec, and cot), use the button or enter [original function] .

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Example

Evaluate . Round appropriately.

Set Mode to Degree

Enter: OR

cot 324.0

: cot 324.0 1.38ANS

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HOWEVER, it is very important to know how to use the reference angle when we are using the inverse trig functions on the calculator to find the angle because the calculator may not directly give you the angle you want.

r-5

y

x

(-12, -5)

-12

Example: Find the value of to the nearest 0.01°

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Examples

Find for 0 360

1) sin 0.418

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Examples

Find for 0 360

2) tan 1.058

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Examples

Find for 0 360

3) cos 0.85

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Examples

Find for 0 360

4) cot 0.012, sin 0

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BONUS PROBLEM

Find for 0 360 without using a calculator.

2sin 1 0

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SUPER DUPER BONUS PROBLEM

Find for 0 360 without using a calculator.

24(sin ) 9 6

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Trig functions of Quadrantal Angles

To find the sine, cosine, tangent, etc. of angles whose terminal

side falls on one of the

axes , we will use the circle.

(..., 180 , 90 , 0 , 90 , 180 , 270 , 360 ,...)

(0, 1) 90

(1, 0)(-1, 0)

(0, -1)

0

270

180

Unit Circle:

Center (0, 0)

radius = 1

x2 + y2 = 1

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Now using the definitions of the trig functions with r = 1, we have:

sin csc1

cos sec1

tan cot

1

1

yy

x

y y r

r y

xy x

x

x

y

x r

r x

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Find the value of the six trig functions for

Example

90

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

270

90

180

sin 901

cos 901

tan 90

1csc 90

1sec 90

cot 90

y y

rx x

ry

xr

y y

r

x xx

y

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Find the value of the six trig functions for

Example

0

sin 0

cos 0

tan 0

1csc 0

1sec 0

cot 0

y

x

y

x

y

xx

y

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Find the value of the six trig functions for

Example

540

sin 540

cos 540

tan 540

csc 540

sec 540

cot 540

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In general, for in radians,

A second way to measure angles is in radians.

Radian Measure (Sect 8.3)

s

r

Definition of Radian:

One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.

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Radian Measure

2 radians corresponds to 360

radians corresponds to 180

radians corresponds to 902

2 6.28

3.14

1.572

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Radian Measure

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Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

180

180

Example

Convert from degrees to radians: 210º

210

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Conversions Between Degrees and Radians

Example

a) Convert from radians to degrees:

b) Convert from radians to degrees: 3.8 (to nearest 0.1°)

3

4

3

4

3.8

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Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):

d) Convert from radians to degrees (exact):13

6

13

6

675

675

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Conversions Between Degrees and Radians

Again!

e) Convert from degrees to radians (to 3 decimal places):

f) Convert from radians to degrees (to nearest tenth): 1 rad

5252

1

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Examples

Find to 4 sig digits for 0 2

sin 0.9540 Hint: There are two answers. Do you remember why?

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Polar Coordinates (Sect 21.10)

A point P in the polar coordinate system is represented by an ordered pair .

• If , then r is the distance of the point from the pole.

is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.

,r

0r

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Polar Coordinates

If , then the point is located units on the ray that extends in the opposite direction of the terminal side of .

0r r

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Example

Plot the point P with polar coordinates 2, .4

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Example

Plot the point with polar coordinates

4,3

4

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5) 3,

3a

) 2,4

b

Plotting Points Using Polar Coordinates

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) 3,0c ) 5,2

d

Plotting Points Using Polar Coordinates

45

A) B)

C) D)

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To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions.

4,3

Example

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Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions.

Example:

Find the polar coordinates (r, θ) for the point for which the rectangular coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

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Conversion from Rectangular Coordinates to Polar Coordinates

If P is a point with rectangular coordinates (x, y), the polar coordinates (r, ) of P are given by

2 2 1tanref

yr x y

x

P

You need to consider the quadrant in

which P lies in order to find the value of .

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) 2, 2a

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

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) 1, 3b

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

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The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)

6: Returns value of given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )

8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).

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End of Section