Half Angle Formulas/Sine/Proof 2
< Half Angle Formulas | Sine
Jump to navigation
Jump to search
Theorem
\(\ds \sin \frac \theta 2\) | \(=\) | \(\ds +\sqrt {\frac {1 - \cos \theta} 2}\) | for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$ | |||||||||||
\(\ds \sin \frac \theta 2\) | \(=\) | \(\ds -\sqrt {\dfrac {1 - \cos \theta} 2}\) | for $\dfrac \theta 2$ in quadrant $\text {III}$ or quadrant $\text {IV}$ |
Proof
Define:
- $u = \dfrac \theta 2$
Then:
\(\ds \sin^2 u\) | \(=\) | \(\ds \frac {1 - \cos 2 u} 2\) | Power Reduction Formulas | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin \frac \theta 2\) | \(=\) | \(\ds \pm \sqrt {\frac {1 - \cos \theta} 2}\) |
We also have that:
- In quadrant $\text I$, and quadrant $\text {II}$, $\sin \theta > 0$
- In quadrant $\text {III}$, and quadrant $\text {IV}$, $\sin \theta < 0$.
$\blacksquare$