Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.4 Trigonometric Functions - Exercises - Page 31: 49


$$ \sin \frac{\theta }{2} =\sqrt{\frac{1}{2} \left( 1-\cos \theta \right)}$$

Work Step by Step

Since \begin{align*} \sin^2\frac{\theta }{2}&= \frac{2\sin^2\frac{\theta }{2}}{2}\\ &= \frac{ \sin^2\frac{\theta }{2}+\sin^2\frac{\theta }{2}}{2}\\ &= \frac{1- \cos^2\frac{\theta }{2}+\sin^2\frac{\theta }{2}}{2}\\ &=\frac{1}{2} \left( 1-\cos \theta \right) \end{align*} Then $$ \sin \frac{\theta }{2} =\sqrt{\frac{1}{2} \left( 1-\cos \theta \right)}$$
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