Double-Angle And Half-Angle Identities Worksheets With Answer Page 5
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6-3 Double-Angle and Half-Angle Identities
2
1
cos
x
2
(1
cos x)
2
sin
x
2
(1
cos x)
2
sin
x
2
(1
cos x)
sin x
2
sin
x
sin x
and
1
cos x
2
(1
cos x)
1
cos x, since
1
cos x is never negative.
All absolute value signs can be dropped, since it can be shown that tan (x/2) and
sin x always have the same sign (a good exercise for you). Thus,
x
sin x
tan
Half-angle identity for tangent
(11)
2
1
cos x
By multiplying the numerator and the denominator in the radicand in equation
(10) by 1
cos x and reasoning as before, we also can obtain
x
1
cos x
tan
Half-angle identity for tangent
(12)
2
sin x
We now list all the half-angle identities for convenient reference.
HALF-ANGLE IDENTITIES
x
1
cos x
sin
2
2
x
1
cos x
cos
2
2
x
1
cos x
sin x
1
cos x
tan
2
1
cos x
1
cos x
sin x
where the sign is determined by the quadrant in which x/2 lies.
(A) Discuss how you would show that, in general,
E x p l o r e / D i s c u s s
x
1
x
1
x
1
2
sin
sin x
cos
cos x
tan
tan x
2
2
2
2
2
2
x
1
(B) Graph y
sin
and y
sin x in the same viewing window.
1
2
2
2
Conclusion? Repeat the process for the other two statements in part A.
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