Half Angle Formula for Tangent/Corollary 2/Proof 2
Jump to navigation
Jump to search
Theorem
- $\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$
Proof
| \(\ds \tan \frac \theta 2\) | \(=\) | \(\ds \frac {\sin \frac \theta 2} {\cos \frac \theta 2}\) | Definition of Real Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin \frac \theta 2} {\cos \frac \theta 2} \frac {2 \sin \frac \theta 2} {2 \sin \frac \theta 2}\) | multiplying both numerator and denominator by $\dfrac {2 \sin \frac \theta 2} {2 \sin \frac \theta 2}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \sin^2 \frac \theta 2} {2 \sin \frac \theta 2 \cos \frac \theta 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \cos \theta} {2 \sin \frac \theta 2 \cos \frac \theta 2}\) | Double Angle Formula for Cosine: Corollary $5$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \cos \theta} {\sin \theta}\) | Double Angle Formula for Sine |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Useful substitutions: Example $1$.