Obj. 16 Trigonometric Ratios (Presentation)

7/30/2019 Obj. 16 Trigonometric Ratios (Presentation)

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Obj. 16 Trigonometric Functions

Unit 5 Trigonometric and Circular Functions


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Concepts and Objectives

Definitions of Trigonometric and Circular Functions

(Obj. #16) Find the values of the six trigonometric functions of

angle .

Find the function values of quadrantal angles.

Identify the quadrant of a given angle.

Find the other function values given one value and

the quadrant


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Trigonometric Ratio Review

In Geometry, we learned that for any given right triangle,

there are special ratios between the sides.

A

opposite

adjacent

=opposite

sin

hypotenuse

A

=adjacent

coshypotenuse

A

=opposite

tanadjacent

A


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Trigonometric Functions

Consider a circle centered at the origin with radius r:

The equation for this circle isx2 +y2 = r2

A point(x,y) on the circle creates a right triangle whose

sides arex,y, and r.

The trig ratios are now (x,y)r

x

y

=siny

r

=cos

x

r

=tany


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Trigonometric Functions

There are three other ratios in addition to the three we

already know : cosecant, secant, and cotangent. These ratios are the inverses of the original three:

(x,y)r

x

y

= =1

csc sin

r

y

= =1

seccos

r

x

= =

1cot tan

x

y


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Finding Function Values

Example: The terminal side of an angle in standard

position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .

8

15

(15, 8)

We know thatx= 15 andy= 8, but

we still have to calculate r:

Now, we can calculate the values.

= +2 2

r x y

= + =2 2

15 8 1717


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Finding Function Values

Example: The terminal side of an angle in standard

position passes through the point(15, 8). Find thevalues of the six trigonometric functions of angle .

8

15

(15, 8)

17

= =8

sin

17

y

r

= =15

cos17

x

r

= =

8tan 15

y

x

= =17

csc

8

r

y

= =17

sec15

r

x

= =

15cot 8

x

y


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The Unit Circle

Angles in standard position whose terminal sides lie on

thex-axis ory-axis (90, 180, 270, etc.) are calledquadrantal angles.

To find function values of quandrantal angles easily, we

Notice that at the quadrantal

angle pointsxandyare either

0, 1, or 1 (ris always 1).

use a circle with a radius of 1, which

is called a unit circle.

90

(0, 1)

(0, 1)

270

180

(1, 0)0/360

(1, 0)


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Values of Quadrantal Angles

Example: Find the values of the six trigonometric

functions for an angle of 270.At 270,x= 0,y= 1, r= 1.

= =

1

sin270 11

= =0

cos270 01

= =

1tan270 undefined

0(0, 1)


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Values of Quadrantal Angles

Example: Find the values of the six trigonometric

functions for an angle of 270.At 270,x= 0,y= 1, r= 1.

= =

1

csc270 11

= =1

sec270 undefined0

= =

0cot 270 0

1

(0, 1)


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Identifying an Angles Quadrant

To identify the quadrant of an angle given certain

conditions, note the following: In the first quadrant,xandyare both positive.

In QII,xis negative andyis positive.

In QIII, both are negative. In QIV,xis positive andyis

IVIII

II I

(+,+)(,+)

(,)

negative.

(+,)


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Identifying an Angles Quadrant

Example: Identify the quadrant (or possible quadrants)

of an angle that satisfies the given conditions.

a) sin > 0, tan < 0 b) cos < 0, sec < 0

I, II II, IV

II

II, III II, III

II, III


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Homework

College Algebra

Page 512: 30-78 (6), 93-102 (3)

Classwork: Algebra & Trigonometry(green book)

Page 728: 77-78

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Obj. 16 Trigonometric Ratios (Presentation)