Chapter 10.5, Problem 3AYU

If $\theta$ is acute, the Half-angle Formula for the sine function is $\sin \frac{\theta }{2}=\_\_\_\_\_\_\_$. (p.506)

To determine

To fill: If $\theta$ is acute, the half angle formula for the sine function is $\text{sin}\left(\frac{\theta }{2}\right)=$ ___________.

Answer to Problem 3AYU

$\text{sin}\frac{\theta }{2}=\surd \frac{1-\cos \theta }{2}$

Explanation of Solution

Given:

$\text{sin}\frac{\theta }{2}$

Formula used:

$2\text{sin}²\left(\theta \right)=1-\text{ cos }2\text{θ}$

Calculation:

Let $\theta =\frac{\alpha }{2}$.

$2\text{sin}²\left(\theta \right)=1-\text{ cos }2\text{θ}$

$\sin \left(\frac{\alpha }{2}\right)=\surd \frac{1-\cos \alpha }{2}$

Replace $\alpha$ by $\theta$.

$\text{sin}\frac{\theta }{2}=\surd \frac{1-\cos \theta }{2}$