Trig 3-6

Trig 3-6

Double-Angle, and Half-Angle Formulas

Multiple Choice

What identity might be helpful in accomplishing this?

• $\cos^{2\ }\theta=1-\sin^2\theta$

• $\sin^2\theta=1-\cos^2\theta$

• $\tan\theta=\frac{\sin\theta}{\cos\theta}$

• $\cos\theta=\frac{1}{\sec\theta}$

Multiple Choice

A Useful Rewrite Continued

Now let’s use that identity to rewrite the right hand side of  $\cos\left(2\theta\right)=\cos^2\theta-\sin^2\theta$   in terms of  $\sin\theta$ . 

•  $\cos\left(2\theta\right)=2\cos^2\theta-1$  

•  $\cos\left(2\theta\right)=1-2\cos^2\theta$  

•  $\cos\left(2\theta\right)=1-2\sin^2\ \theta$  

•  $\cos\left(2\theta\right)=2\sin^2\theta-1$  

Multiple Choice

Quick Check 1

If  $\sin\theta=\frac{4}{5}$  and  $\theta$   lies in quadrant I, find the exact value of  $\sin2\theta$ . 

•  $\frac{8}{5}$  

•  $\frac{24}{25\ }$  

•  $-\frac{25}{25}$  

•  $\frac{2}{5}$  

Multiple Choice

Quick Check 2

Find the exact value of  $\cos\left(75\degree\right)$ using a half-angle identity.

•  $\frac{\sqrt{2-\sqrt{3}}}{2}$  

•  $\sqrt{\frac{1-\sqrt{3}}{2}}$  

• 1

•  $\sqrt{2-\sqrt{3}}$  

Multiple Choice

Quick Check 3

If  $\cos\alpha=-\frac{3}{5}$ ,  $π<α<3π/2\$ , find the exact value of  $\cos\frac{\alpha}{2}$  

Hint: Use what you already know from the previous slide! 

•  $-\frac{3}{10}$  

•  $-\frac{\sqrt{5}}{5}$  

•  $1$  

•  $\frac{\sqrt{5}}{5}$