# Double-Angle And Half-Angle Identities Worksheets With Answer Page 6

476
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
E X A M P L E
Finding Exact Values
3
Compute the exact value of sin 165° without a calculator using a half-angle
identity.
330°
sin 165°
sin
S o l u t i o n
2
Use half-angle identity for sine
1
cos 330°
with a positive radical, since sin 165°
2
is positive.
1
( 3/2)
2
2
3
2
Compute the exact value of tan 105° without a calculator using a half-angle
M A T C H E D P R O B L E M
3
identity.
E X A M P L E
Finding Exact Values
4
Find the exact values of cos (x/2) and cot (x/2) without using a calculator if
3
sin x
,
x
3 /2.
5
Draw a reference triangle in the third quadrant, and ﬁnd cos x. Then use appro-
S o l u t i o n
priate half-angle identities.
2
2
a
5
( 3)
4
a
4
cos x
5
3
5
(a,
3)
If
x
3 /2, then
x
3
Divide each member of
x
3 /2 by 2.
2
2
4
Thus, x/2 is an angle in the second quadrant where cosine and cotangent are neg-
ative, and
x
1
cos x
x
1
sin x
cos
cot
2
2
2
tan (x/2)
1
cos x
4
3
1
(
)
5
5
1
4
3
2
1
(
)
5
1
10
or
10
10