9.24 Double-angle identities

Double-angle identities

In this lesson, we will learn how to make use of the double-angle identities, a.k.a. double-angle formulas to find the sine and cosine of a double angle. It’s hard to simplify complex trigonometric functions without these formulas.


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sin2θ=2sinxcosx \sin 2\theta = 2\sin x\cos x
cos2θ=cos2xsin2x \cos 2 \theta = {\cos ^2}x - {\sin^2}x
=2cos2x1 = \;2{ \cos ^2}x - 1
=12sin2x = \;1 - 2{\sin ^2}x
tan2θ=2tanx1tan2x \tan 2 \theta = {{2\tan x} \over {1 - \tan ^2}x}
  • 1.
    Express each of the following in terms of a single trigonometric function:
  • 2.
    Prove identities
    • a)
      sin2x \sin 2x
    • b)
      sec2x\sec 2x
    • c)
      tan2x\tan 2x
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Double-angle identities

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