EXAMPLE 1 Find trigonometric values

Given that sin = and < < π, find the values of the other five trigonometric functions of .

45

π2

EXAMPLE 1 Find trigonometric values

SOLUTION

STEP 1 Find cos .

Write Pythagorean Identity.sin + cos 2 2 = 1

Substitute for sin .45

( ) + cos 45

2 21=

Subtract ( ) from each side.45

2cos 2 24

51 – ( )=

Simplify.cos 2 925=

Take square roots of each side.cos 35

+–=

Because is in Quadrant II, cos is negative.

cos 35

–=

EXAMPLE 1 Find trigonometric values

STEP 2 Find the values of the other four trigonometric functions of using theknown values of sin and cos .

tan sin cos = =

4535

–= 4

3–

cot cos sin = =

45

35

–= 3

4–

EXAMPLE 1 Find trigonometric values

csc sin = =

145

= 54

sec cos = =

35

–

1 =53

–

EXAMPLE 2 Simplify a trigonometric expression

Simplify the expression tan ( – ) sin .π2

Cofunction Identitytan ( – ) sin π2

cot sin =

Cotangent Identity= ( ) ( sin )cos sin

Simplify.= cos

EXAMPLE 3 Simplify a trigonometric expression

2Simplify the expression csc cot + .sin

Reciprocal Identity2csc cot +

sin csc cot + csc 2=

Pythagorean Identity= csc (csc – 1) + csc 2

Distributive property= csc – csc + csc 3

Simplify.= csc 3

GUIDED PRACTICE for Examples 1, 2, and 3

Find the values of the other five trigonometric functions of .

16

1. cos , 0 < <= π2

SOLUTION

sin = 356

sec = 6

csc

cot

= 6 35

35

= 3535

GUIDED PRACTICE for Examples 1, 2, and 3

Find the values of the other five trigonometric functions of .

2. sin = , π <

3π 2

–3 7

SOLUTION

cos –= 2 10 7

tan =20

3 10

csc = 73

–

sec = – 720

10

cot 2 10 3

= –

GUIDED PRACTICE for Examples 1, 2, and 3

3. sin x cot x sec x

Simplify the expression.

1ANSWER

4. tan x csc xsec x

1ANSWER

cos –1 π2

–

1 + sin (– )5.

–1ANSWER