Double-Angle And Half-Angle Identities Worksheets With Answer Page 3

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473
6-3 Double-Angle and Half-Angle Identities
We start with the right side:
V e r i f i c a t i o n
2
sin
x
1
2
2
1
tan
x
cos
x
Quotient identities
2
2
1
tan
x
sin
x
1
2
cos
x
2
2
cos
x
sin
x
Algebra
2
2
cos
x
sin
x
2
2
cos
x
sin
x
Pythagorean identity
Double-angle identity
cos 2x
Key Algebraic Steps in Example 1
2
2
a
a
2
1
b
1
2
2
2
2
b
b
b
a
2
2
2
2
a
a
b
a
2
1
b
1
2
2
b
b
2 tan x
Verify the identity sin 2x
.
M A T C H E D P R O B L E M
2
1
tan
x
1
E X A M P L E
Finding Exact Values
2
Find the exact values, without using a calculator, of sin 2x and cos 2x if
3
tan
x
and x is a quadrant IV angle.
4
First draw the reference triangle for x and find any unknown sides:
S o l u t i o n
2
2
r
( 3)
4
5
4
3
sin x
5
4
cos x
5
3
r
Now use double-angle identities for sine and cosine:
3
4
24
sin 2x
2 sin x cos x
2(
)(
)
5
5
25
4
7
2
2
cos 2x
2 cos
x
1
2(
)
1
5
25
4
Find the exact values, without using a calculator, of cos 2x and tan 2x if sin
x
M A T C H E D P R O B L E M
5
2
and x is a quadrant II angle.

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