20.3 Double-Angle Formulas

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

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:

\begin{array}{rcl} \sin (α + α) & = & \sin (2 α) = \sin α \cos α + \cos α \sin α = 2 \sin α \cos α \\ \cos (α + α) & = & \cos α \cos α - \sin α \sin α = \cos^{2} α - \sin^{2} α \\ \tan (α + α) & = & \frac{\tan α + \tan α}{1 - \tan α \tan α} = \frac{2 \tan α}{1 - \tan^{2} α} \end{array}

Then using the basic identity $\sin^{2} x + \cos^{2} x = 1,$ other forms of the equation for $\cos 2 α$ may be derived. Summarizing these forms, we have

\sin 2 α = 2 \sin α \cos α

(20.21)

\cos 2 α = \cos^{2} α - \sin^{2} α

(20.22)

= 2 \cos^{2} α - 1

(20.23)

= 1 - 2 \sin^{2} α

(20.24)

\tan 2 α = \frac{2 \tan α}{1 - \tan^{2} α}

(20.25)

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

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Basic Technical Mathematics, 11th Edition by Allyn J. Washington, Richard Evans

20.3 Double-Angle Formulas

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