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5.3 Trigonometric Functions of Any Angles.The Unit Circle

Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu

1. Use the definitions of trigonometric functions of any angle.2. Use the signs of the trigonometric functions.3. Find reference angles.4. Use reference angles to evaluate trigonometric functions.

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Definitions of Trigonometric Functions of Any Angle

Let be any angle in standard position and let P = (x, y) be a point on the terminal side of If is the distance from (0, 0) to (x, y), the six trigonometric functions of are defined by the following ratios:

2 2r x y

sinyr

cosxr

tan , 0yx

x

csc , 0ry

y

sec , 0rx

x

cot , 0xy

y

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Example: Evaluating Trigonometric Functions

Let P = (1, –3) be a point on the terminal side of Find each of the six trigonometric functions of P = (1, –3) is a point on the terminal side of x = 1 and y = –3

2 2r x y 2 2(1) ( 3) 1 9 10

sinyr

3

10

3 10 3 101010 10

1

10cos

xr

1 10 10

1010 10

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Example: Evaluating Trigonometric Functions (continued)

Let P = (1, –3) be a point on the terminal side of Find each of the six trigonometric functions of

We have found that

10.r

tanyx

3

31

cscry

103

secrx

1010

1

cotxy

13

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Example: Evaluating Trigonometric Functions (continued)

Let P = (1, –3) be a point on the terminal side of Find each of the six trigonometric functions of

3 10sin

10

10cos

10

tan 3

10csc

3

sec 10

1cot

3

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Example: Trigonometric Functions of Quadrantal Angles

Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:

If then the terminal side of the angle is on the positive x-axis. Let us select the point P = (1, 0) with x = 1 and y = 0.

0 0

0 0 radians,

cosxr

11

1

cscry

10

is undefined.csc

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Example: Trigonometric Functions of Quadrantal Angles

Evaluate, if possible, the cosine function and the cosecant

function at the following quadrantal angle:

If then the terminal side of the angle is

on the positive y-axis. Let us select the point P = (0, 1) with

x = 0 and y = 1.

902

90 radians,2

cosxr

00

1

cscry

11

1

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Example: Trigonometric Functions of Quadrantal Angles

Evaluate, if possible, the cosine function and the cosecant function at the following quadrantal angle:

If then the terminal side of the angle is on the positive x-axis. Let us select the point P = (–1, 0) with x = –1 and y = 0.

180 180 radians,

cosxr

11

1

cscry

10

csc is undefined.

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Example: Trigonometric Functions of Quadrantal Angles

Evaluate, if possible, the cosine function and the

cosecant function at the following quadrantal angle:

If then the terminal side

of the angle is on the negative y-axis. Let us select

the point P = (0, –1) with x = 0 and y = –1.

3270

2

3270 radians,

2

cosxr

00

1

cscry

11

1

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The Signs of the Trigonometric Functions

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Example: Finding the Quadrant in Which an Angle Lies

If name the quadrant in which the angle lies.

sin and cos 0,

lies in Quadrant III.

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Example: Evaluating Trigonometric Functions

Given find

Because both the tangent and the cosine are

negative, lies in Quadrant II.

1tan and cos 0,

3 sin and sec .

tanyx

13

3, 1x y

2 2r x y 2 2( 3) (1) 9 1 10

sinyr

1 10 101010 10

secrx

10 103 3

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Definition of a Reference Angle

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Example: Finding Reference Angles

Find the reference angle, for each of the following angles:

a.

b.

c.

d.

210

74

240

3.6

180 210 180 30

2 72

4 8 7

4 4 4

60

3.6 3.14 0.46

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Finding Reference Angles for Angles Greater Than 360° or Less Than –360°(2 ) ( 2 )

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Example: Finding Reference Angles

Find the reference angle for each of the following angles:

a.

b.

c.

665

154

113

360 305 55 7 8 7

24 4 4 4

11 123 3 3

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Using Reference Angles to Evaluate Trigonometric Functions

A Procedure for using reference Angles to Evaluate Trigonometric Functions

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Example: Using Reference Angles to Evaluate Trigonometric Functions Use reference angles to find the exact value of

Step 1 Find the reference angle, and

Step 2 Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.

sin135 .

sin

360 360 300 60

sin300 sin 60 32

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Example: Using Reference Angles to Evaluate Trigonometric Functions

Use reference angles to find the exact value of Step 1 Find the reference angle, and

Step 2 Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.

5tan .

4

tan

5 44 4 4

5tan tan

4 4 1

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Example: Using Reference Angles to Evaluate Trigonometric Functions

Use reference angles to find the exact value of

Step 1 Find the reference angle, and

Step 2 Use the quadrant in which lies to prefix the appropriate sign to the function value in step 1.

sec .6

sec .

sec sec6 6

2 3

3