Double-Angle And Half-Angle Identities Worksheets With Answer Page 4
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6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
Half-Angle Identities
Half-angle identities are simply double-angle identities stated in an alternate form.
Let’s start with the double-angle identity for cosine in the form
2
cos 2m
1
2 sin
m
Now replace m with x/2 and solve for sin (x/2) [if 2m is twice m, then m is half
of 2m—think about this]:
x
2
cos x
1
2 sin
2
x
1
cos x
2
sin
2
2
x
1
cos x
sin
Half-angle identity for sine
(7)
2
2
where the choice of the sign is determined by the quadrant in which x/2 lies.
To obtain a half-angle identity for cosine, start with the double-angle identity
for cosine in the form
2
cos 2m
2 cos
m
1
and let m
x/2 to obtain
x
1
cos x
cos
Half-angle identity for cosine
(8)
2
2
where the sign is determined by the quadrant in which x/2 lies.
To obtain a half-angle identity for tangent, use the quotient identity and the
half-angle formulas for sine and cosine:
x
1
cos x
sin
x
2
2
1
cos x
tan
2
x
1
cos x
1
cos x
cos
2
2
Thus,
x
1
cos x
tan
Half-angle identity for tangent
(9)
2
1
cos x
where the sign is determined by the quadrant in which x/2 lies.
Simpler versions of equation (9) can be obtained as follows:
x
1
cos x
tan
(10)
2
1
cos x
1
cos x
1
cos x
1
cos x
1
cos x
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