20.3 Double-Angle Formulas

  • Formula for sin 2α • Formulas for cos 2α • Formula for tan 2α

If we let β = α in the sum formulas for sine, cosine, and tangent (given in Section 20.2), we can derive the important double-angle formulas:

sin(α + α) = sin(2α) = sin α cos α + cos α sin α = 2 sin α cos αcos(α + α) = cos α cos α − sin α sin α = cos2 α − sin2 αtan(α + α) = tan α + tan α1 − tan α tan α = 2 tan α1 − tan2 α

Then using the basic identity sin2 x + cos2 x = 1 ,  other forms of the equation for cos 2α may be derived. Summarizing these forms, we have

sin 2α = 2 sin α cos α (20.21)
cos 2α = cos2 α − sin2 α (20.22)
 = 2 cos2 α − 1 (20.23)
 = 1 − 2 sin2 α (20.24)
tan 2α = 2 tan α1 − tan2 α (20.25)

These double-angle formulas are widely used in ...

Get Basic Technical Mathematics, 11th Edition now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.