Double Angle Formulas for Sine and Cosine
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Theorem
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- $\sin \left({2 \theta}\right) = 2 \sin \theta \cos \theta$
- $\cos \left({2 \theta}\right) = \cos^2 \theta - \sin^2 \theta$
- $\displaystyle \tan \left({2 \theta}\right) = \frac {2\tan \theta} {1 - \tan^2 \theta}$
where $\sin, \cos, \tan$ are sine, cosine and tangent.
Corollary
- $\cos \left({2 \theta}\right) = 2 \ \cos^2 \theta - 1$
- $\cos \left({2 \theta}\right) = 1 - 2 \ \sin^2 \theta$
Proof 1
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \cos \left({2 \theta}\right) + i \sin \left({2 \theta}\right)\) | \(=\) | \(\displaystyle (\cos\theta + i\sin\theta)^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | De Moivre's Formula | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \cos^2\theta + i^2\sin^2\theta + 2i\cos\theta\sin\theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \cos^2\theta - \sin^2\theta + 2i\cos\theta\sin\theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Equate real and imaginary parts:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \cos \left({2 \theta}\right)\) | \(=\) | \(\displaystyle \cos^2 \theta - \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | (real parts) | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \sin \left({2 \theta}\right)\) | \(=\) | \(\displaystyle 2 \cos \theta \sin \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | (imaginary parts) |
Furthermore, computing as follows:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \tan \left({2 \theta}\right)\) | \(=\) | \(\displaystyle \frac {\sin \left({2 \theta}\right)} {\cos \left({2 \theta}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of tangent | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {2 \cos \theta \sin \theta} {\cos^2 \theta - \sin^2 \theta}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Preceding equations | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\frac {2 \cos \theta \sin \theta} {\cos^2 \theta} } {\frac {\cos^2 \theta - \sin^2 \theta} {\cos^2 \theta} }\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {2 \tan \theta} {1 - \tan^2 \theta}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Simplifying; definition of tangent |
$\blacksquare$
Proof 2
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \sin \left({2\theta}\right)\) | \(=\) | \(\displaystyle \sin \left({\theta + \theta}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sin \theta \cos \theta + \cos \theta \sin \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sine and Cosine of Sum | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle 2 \sin \theta \cos \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \cos \left({2\theta}\right)\) | \(=\) | \(\displaystyle \cos \left({\theta + \theta}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \cos \theta \cos \theta - \sin \theta \sin \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sine and Cosine of Sum | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \cos^2 \theta - \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \tan \left({2\theta}\right)\) | \(=\) | \(\displaystyle \tan \left({\theta + \theta}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\tan \theta + \tan \theta} {1 - \tan \theta \tan \theta}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Tangent of Sum | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {2\tan \theta} {1 - \tan^2 \theta}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
$\blacksquare$
Proof of Corollary
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \cos \left({2\theta}\right)\) | \(=\) | \(\displaystyle \cos^2 \theta - \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Main result | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \cos^2 \theta - \left({1 - \cos^2 \theta}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle 2 \ \cos^2\theta - 1\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \cos \left({2\theta}\right)\) | \(=\) | \(\displaystyle \cos^2 \theta - \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Main result | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \left({1 - \sin^2 \theta}\right) - \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle 1 - 2 \ \sin^2 \theta\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
$\blacksquare$
Sources
- Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $5.38, \ 5.39, \ 5.40$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.3 \ (3) \ \text{(iii), (iv)}$
