Double-Angle And Half-Angle Identities Worksheets With Answer
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6-3 Double-Angle and Half-Angle Identities
Section 6-3 Double-Angle and Half-Angle Identities
Double-Angle Identities
Half-Angle Identities
This section develops another important set of identities called double-angle and
half-angle identities. We can derive these identities directly from the sum and dif-
ference identities given in Section 6-2. Even though the names use the word
“angle,” the new identities hold for real numbers as well.
Double-Angle Identities
Start with the sum identity for sine,
sin (x
y)
sin x cos y
cos x sin y
and replace y with x to obtain
sin (x
x)
sin x cos x
cos x sin x
On simplification, this gives
sin 2x
2 sin x cos x
Double-angle identity for sine
(1)
If we start with the sum identity for cosine,
cos (x
y)
cos x cos y
sin x sin y
and replace y with x, we obtain
cos (x
x)
cos x cos x
sin x sin x
On simplification, this gives
2
2
cos 2x
cos
x
sin
x
First double-angle identity for cosine
(2)
Now, using the Pythagorean identity
2
2
sin
x
cos
x
1
(3)
in the form
2
2
cos
x
1
sin
x
(4)
and substituting it into equation (2), we get
2
2
cos 2x
1
sin
x
sin
x
On simplification, this gives
2
cos 2x
1
2 sin
x
Second double-angle identity for cosine
(5)
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