Section 6.3 Trigonometric Functions of Angles

6.3 - Trigonometric Functions of Angles

Section 6.3 Trigonometric Functions of

Angles

Chapter 6 – Trigonometric Functions: Right Triangle Approach


Let POQ be a right triangle with an acute angle where is in standard position. The point P is on the terminal side of .

P(x, y)

y

x

r

QOx

y

2 2r x y


Definition of Trigonometric Functions

Let be an angle in standard position and let P(x, y) be a point on the terminal side. If is the distance from the origin to the point P(x, y), then

2 2r x y

sin

cos

tan 0

y

rx

ry

xx

csc 0

sec 0

cot 0

ry

y

rx

xx

yy


Quadrantal AnglesQuadrantal Angles are angles that are coterminal with

the coordinate axis.

5.2 - Trigonometric Function of Real Numbers

Remember:Signs of the Trig Functions

Quad Positive Functions Negative Functions I All None II sin, csc cos, sec, tan,

cot III tan, cot sin, csc, cos,

secIV cos, sec sin, csc, tan, cot


Examples – pg. 460Find the quadrant in which lies from the

information given. 35. sin 0 and cos 0

36. tan 0 and sin 0

37. sec 0 and tan 0

38. csc 0 and cos 0

5.1 - The Unit Circle

Reference NumberLet be an angle in standard position. The

reference angle` associated with is the acute angle formed by the terminal side of and the x-axis.

5.1 - The Unit Circle

Evaluating Trig Functions for Any Angle

To find the values of the trigonometric functions for any angle , we carry out use the following steps:

1. Find the reference angle` associated with the angle .

2. Determine the sign of the trigonometric function of by noting the quadrant in which lies.

3. The value of the trigonometric function of is the same, except possibly for sign, as the value of the trigonometric function of`.


Examples – pg. 459Find the reference angle for the given angle.

3. (a) 150 (b) 330 (c) 30

11 11 117. (a) (b) (c)

4 6 3


Examples – pg. 459Find the exact value of the trigonometric function.

11. sin150 19. cos570

213. cos 210 23. sin

3

317. csc 630 25. sin

2

5.2 - Trigonometric Function of Real Numbers

Fundamental Identities

Reciprocal Identities

Pythagorean Identities

1 1csc sec

sin cos

sin 1 costan cot =

cos tan sin

t tt t

t tt t

t t t

2 2 2 2 2 2sin cos 1 tan 1 sec 1+cot csct t t t t t


Examples – pg. 460Write the first trigonometric function in terms of the

second for in the given quadrant.

39. tan , cos ; in Quad III

40. cot , sin ; in Quad II

41. cos , sin ; in Quad IV

42. sec , sin ; in Quad I

43. sec , tan ;

in Quad II

44. csc , cot ; in Quad III


Examples – pg. 460Find the given values of the trigonometric functions

of from the information given.

345. sin , in Quadrant II

5

347. tan , cos 0

4


Area of a TriangleThe area A of a triangle with sides of length a and b

with included angle is

1sin

2ab A


Examples – pg. 460

54. Find the area of a triangle with sides of length 7 and 9 and

included angle 72 .

55. Find the area of a triangle with sides of length 10 and 22 and

included angle 10 .


Examples – pg. 460

2

56. Find the area of an equilateral triangle with side length 10.

57. A triangle has an area of 16 in , and two of the sides of the

triangles have lengths 5 in. and 7 in. Find the angle inclu

2

ded

by these two sides.

58. An isosceles triangle has an area of 24 cm , and the angle between

5 the two equal sides is . What is the length of the two equal sides?

6

Section 6.3 Trigonometric Functions of Angles

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