# Double-Angle And Half-Angle Identities Worksheets With Answer Page 7

477
6-3 Double-Angle and Half-Angle Identities
Find the exact values of sin (x/2) and tan (x/2) without using a calculator if
M A T C H E D P R O B L E M
4
4
cot x
,
/2
x
.
3
E X A M P L E
Identity Verification
x
tan x
sin x
5
2
Verify the identity:
sin
2
2 tan x
x
1
cos x
Half-angle identity for sine
sin
V e r i f i c a t i o n
2
2
x
1
cos x
2
sin
Square both sides.
2
2
tan x
1
cos x
Algebra
tan x
2
tan x
tan x cos x
Algebra
2 tan x
tan x
sin x
Quotient identity
2 tan x
x
tan x
sin x
2
Verify the identity
cos
.
M A T C H E D P R O B L E M
2
2 tan x
5
A n s w e r s t o M a t c h e d P r o b l e m s
sin x
sin x
2
2
cos
x 2
2 tan x
cos x
cos x
2 sin x cos x
1.
2 sin x cos x
sin 2x
2
2
2
2
2
1
tan
x
sin
x
sin
x
cos
x
sin
x
2
1
cos
x 1
2
2
cos
x
cos
x
7
24
2.
cos 2x
, tan 2x
3.
3
2
4.
sin (x/2)
3 10/10, tan (x/2)
3
25
7
x
1
cos x
tan x
1
cos x
tan x
tan x cos x
tan x
sin x
2
5. cos
2
2
tan x
2
2 tan x
2 tan x
EXERCISE 6-3
x
1
cos x
5.
sin
, x
2
2
(Choose the correct sign.)
A
In Problems 1–6, verify each identity for the values indicated.
x
1
cos x
6.
cos
, x
2
2
2
2
2
1. cos 2x
cos
x
sin
x, x
30°
(Choose the correct sign.)
2. sin 2x
2 sin x cos x, x
45°
In Problems 7–10, ﬁnd the exact value without a calculator
2
3.
tan 2x
, x
using double-angle and half-angle identities.
cot x
tan x
3
7. sin 22.5°
8. tan 75°
2 tan x
4. tan 2x
, x
2
1
tan
x
6
9. cos 67.5°
10. tan 15°