Question

I need help with this practice problem It’s from my trigonometry prep guide

Answer 1

Reference angles

1.

$\begin{gathered} -\frac{23\pi}{6}\cdot\frac{180}{\pi}=-690 \\ -690+720=30\text{ is in Q1} \\ \theta_{ref}=30 \\ 30\cdot\frac{\pi}{180}=\frac{\pi}{6} \end{gathered}$

$\theta=-\frac{23\pi}{6}\rightarrow\theta_{ref}=\frac{\pi}{6}$

2.

$\begin{gathered} \frac{7}{5}\pi\cdot\frac{180}{\pi}=252 \\ 252\text{ is in Q}3 \\ \theta_{ref}=252-180=72 \\ 72\cdot\frac{\pi}{180}=\frac{2}{5}\pi \end{gathered}$$\theta=\frac{7}{5}\pi\rightarrow\theta_{ref}=\frac{2}{5}\pi$

3.

$\begin{gathered} -\frac{11\pi}{12}\cdot\frac{180}{\pi}=-165 \\ -165+360=195 \\ 195\text{ is in Q3} \\ \theta_{ref}=195-180=15 \\ 15\cdot\frac{\pi}{180}=\frac{\pi}{12} \end{gathered}$$\theta=-\frac{11\pi}{12}\rightarrow\theta_{ref}=\frac{\pi}{12}$