Section 6.3Properties of the

Trigonometric Functions

If sin > 0 and cos < 0, name the quadrant in which the angle lies.

For sin > 0 the y value must be positive so the angle must be in quadrant I or II.

For cos < 0 the x value must be negative so the angle must be in quadrant II or III.

Therefore, this angle must lie in quadrant II.

10 3 10Given sin and cos , find the value of each of the four remaining 10 10

trigonometric functions of .

10sin 10 10 110tancos 10 33 10 3 10

10

1 1cot 31tan

3

1 1 10csc 10sin 1 1

100 0

1 1 10 10sec

cos 33 10 3 1010

22

Find the exact value of each expression. Do not use a calculator.

cos1 3(a) cos 35 (b) cotcsc 35 3sin

3

2 2 22

1(a) cos 35 sin 35 cos 35 1csc 35

cos3(b) cot cot cot 0

3 3 3sin3

2Given that sin and cos , find the exact value of each 5

of the remaining five trigonometric functions of .

21cos = 5

xr

5csc = 2

ry

5 5 21sec = 2121

rx

21cot = 2

xy

θP(x,2)

r = 5

2 2 21tan = 2121

yx

2Since sin 0 and cos 0, is in quadrant II.5

2 2 2The circle has a radius of 5 so its equation is 5x y

2 2 2 is a point on the circle so 5 2P x

21x 2 21x

2Given that sin and is an acute angle, find the exact value of each 5

of the remaining five trigonometric functions of .

2 2sin c 1os 2

22 co 1s5

2 4 2112 2

cos5 5

21 212 5

cos5

Since θ is in quadrant II, x values are negative

2sin 2 55tancos 521 21

5

2 2121

1 21cottan 2

1 1 5csc

2sin 25

1 1 5 5 21seccos 2121 21

5

P(-1,-3)

θ

1Given that cot and sin , find the exact value of each 3

of the remaining five trigonometric functions of .

1 10cos = 1010

xr

3tan = 31

yx

10csc = 3

ry

10sec = 101

rx

3 3 10sin = 1010

yr

1Since cot 0 and sin 0, is in quadrant III.3

2 2 2The circle has a radius of 3 so its equation is 3x y

2 is a point on the circle so 1 9 10P r

10r

2 2cot 1 csc 2

21 1 csc3

2 1csc9

1019

10 109 3

csc

Since sin θ < 0, csc θ is negative.

sin 3 10 110costan 3

3 1010

110 3 0

1 1tan 31cot3

1 1 3 3 10sincsc 1010 10

3

1 1 10sec 10cos 10 1

100

1Given that cot and sin , find the exact value of each 3

of the remaining five trigonometric functions of .

Find the exact value of:

(a) cos (60°) (b) sin (390°) (c) tan 374

1(a) cos 60 cos602

1(b) sin 390 sin 390 sin 30 360 sin 302

37 37 36(c) tan tan tan4 4 4 4

tan 9 tan 14 4