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Copyright © 2009 Pearson Addison-Wesley 1.1-1 1.3-1

Trigonometric Functions 1.3 • Define the six trig functions for an angle.

• Find the values of trig functions given an angle, a point on the terminal side of the angle, or one of the trig function values.

Copyright © 2009 Pearson Addison-Wesley 1.3-2

Trigonometric Functions

Let (x, y) be a point other the origin on the terminal

side of an angle in standard position. The

distance from the point to the origin is

𝒓 = 𝒙𝟐 + 𝒚𝟐.

(Pythagorean

Theorem)

*r is always positive.


The six trigonometric functions are all the

possible ratios of the three sides of a right

triangle.


Sides x y r



The six trigonometric functions of θ are defined as follows:

sin 𝜃 =𝑦

𝑟 cos 𝜃 =

𝑥

𝑟 tan 𝜃 =

𝑦

𝑥

csc 𝜃 =𝑟

𝑦 sec 𝜃 =

𝑟

𝑥 cot 𝜃 =

𝑥

𝑦

RECIPROCALS


The terminal side of angle in standard position

passes through the point (8, 15). Find the values of

the six trigonometric functions of angle .

Example 1 FINDING FUNCTION VALUES GIVEN A

POINT ON THE TERMINAL SIDE OF Ө

𝑥 = 8 𝑦 = 15 𝑟 = 𝑥2 + 𝑦2

𝑟 = 82 + 152

𝑟 = 64 + 225

𝑟 = 289 𝑟 = 17


Example 1 FINDING FUNCTION VALUES GIVEN A

POINT ON THE TERMINAL SIDE OF Ө

(continued)

𝑥 = 8, 𝑦 = 15, 𝑟 = 17

Use the definitions of the trigonometric functions.

sin 𝜃 =𝑦

𝑟=

15

17 cos 𝜃 =

𝑥

𝑟=

8

17 tan 𝜃 =

𝑦

𝑥=

15

8

csc 𝜃 =𝑟

𝑦=

17

15 sec 𝜃 =

𝑟

𝑥=

17

8 cot 𝜃 =

𝑥

𝑦=

8

15


The terminal side of angle in standard position

passes through the point (–3, –4). Find the values

of the six trigonometric functions of angle .

Example 2 FINDING FUNCTION VALUES OF AN

ANGLE

𝑥 = −3 𝑦 = −4 𝑟 = 𝑥2 + 𝑦2

𝑟 = (−3)2+(−4)2

𝑟 = 9 + 16

𝑟 = 25 𝑟 = 5



ANGLE (continued)

𝑥 = −3, 𝑦 = −4, 𝑟 = 5

Use the definitions of the trigonometric functions.

sin 𝜃 =−4

5 cos 𝜃 =

−3

5 tan 𝜃 =

4

3

csc 𝜃 =5

−4 sec 𝜃 =

5

−3 cot 𝜃 =

3

4



ANGLE GIVEN THE EQUATION OF THE

TERMINAL SIDE

Find the six trigonometric function values of the

angle θ in standard position, if the terminal side of θ

is defined by x + 2y = 0, x ≥ 0.

1. Put the equation in slope-intercept form, y = mx + b and graph.

𝒚 = −𝟏

𝟐𝒙 + 𝟎

2. Choose a point on the line.

(𝟐, −𝟏)

𝜽




TERMINAL SIDE (continued)

𝑥 = 2 𝑦 = −1

𝑟 = 22 + (−1)2= 5

sin 𝜃 =−1

5∙

5

5= −

5

5 csc 𝜃 = − 5

cos 𝜃 =2

5∙

5

5=

2 5

5 sec 𝜃 =

5

2

tan 𝜃 = −1

2 cot 𝜃 = −2




TERMINAL SIDE

Find the six trigonometric function values of the

angle θ in standard position, if the terminal side of θ

is defined by 6x – 5y = 0, x ≤ 0.

𝑦 =6

5𝑥 + 0

(−5, −6) 𝜽




TERMINAL SIDE (continued)

𝑥 = −5 𝑦 = −6

𝑟 = (−5)2+(−6)2= 61

sin 𝜃 =−6

61∙

61

61= −

6 61

61 csc 𝜃 = −

61

6

cos 𝜃 =−5

61∙

61

61= −

5 61

61 sec 𝜃 = −

61

5

tan 𝜃 =6

5 cot 𝜃 =

5

6



ANGLE

Find the following values for −135°. Let 𝑡 = 1, so −1, −1

𝑟 = −1 2 + −1 2 = 2

sin −135° =−1

2∙

2

2= −

2

2

cos −135° =−1

2∙

2

2= −

2

2

tan −135° =−1

−1= 1

(−𝑡, −𝑡)

45°

45°

𝑡

𝑡



ANGLE

Find the following values for 510°.

Let 𝑡 = 1, so − 3, 1

𝑟 = − 32

+ 12 = 2

sin 510° =1

2

cos 510° = −3

2

tan 510° =1

− 3∙

3

3= −

3

3

(−𝑡 3, 𝑡)

60°

30° 𝑡

𝑡 3


Signs of Function Values

IV

III

II

I

csc sec cot tan cos sin

in

Quadrant

+ + + + + +

+ +

+ +

+ +

− − − −

− − − −

− − − −


Ranges of Function Values

All Students Take Calculus

A S

T C


Determine the signs of the trigonometric functions of

an angle in standard position with the given measure.

(a) 87° QI All positive

(b) 300° QIV cos & sec are positive

sin, csc, tan, cot are negative

(a) –200° QII sin & csc are positive

cos, sec, tan, cot are negative

Example 7 DETERMINING SIGNS OF FUNCTIONS

OF NONQUADRANTAL ANGLES


Identify the quadrant (or quadrants) of any angle

that satisfies the given conditions.

Example 8 IDENTIFYING THE QUADRANT OF AN

ANGLE

(a) sin > 0, tan < 0. QII

(b) cos < 0, sec < 0 QII, QIII



ANGLE

Find the remaining trigonometric functions of 𝜃 if

cos 𝜃 = −12

13 and 𝜃 terminates in QII.

𝑥 = −12 sin 𝜃 =5

13 csc 𝜃 =

13

5

𝑟 = 13

𝑥2 + 𝑦2 = 𝑟2 sec 𝜃 = −13

12

(−12)2+𝑦2 = 132

𝑦2 = 25 tan 𝜃 = −5

12 cot 𝜃 = −

12

5

𝑦 = 5



ANGLE

Find the remaining trigonometric functions of 𝜃 if

csc 𝜃 = −3 and cos 𝜃 < 0.

𝑦 = −1 sin 𝜃 = −1

3

𝑟 = 3

𝑥2 + −1 2 = 32 cos 𝜃 = −2 2

3 sec 𝜃 =

3

−2 2∙

2

2= −

3 2

4

𝑥2 = 8

𝑥 = −2 2 tan 𝜃 =−1

−2 2∙

2

2=

2

4 cot 𝜃 = 2 2


Caution

In problems like in Examples 9 and

10, be careful to choose the correct

sign based on the quadrants.

Independent Practice

1.3 WS


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