EXAMPLE 1 Find trigonometric values Given that sin = and < < π, find the values of the other five...
Transcript of EXAMPLE 1 Find trigonometric values Given that sin = and < < π, find the values of the other five...
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