Question

homework assignment #7.4
score: 3/20 3/20 answered
question 4
for each angle \\( \theta \\) in standard position, find its exact reference angle in radians.
(i) if \\( \theta = \frac{ -7\pi }{ 6 } \\) then \\( \hat{\theta} = \\)
(ii) if \\( \theta = \frac{ 5\pi }{ 3 } \\) then \\( \hat{\theta} = \\)
(iii) if \\( \theta = \frac{ -5\pi }{ 4 } \\) then \\( \hat{\theta} = \\)

Explanation:

Step1: Adjust negative angle to positive coterminal

Add $2\pi$ to $-\frac{7\pi}{6}$: $-\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}$
$\frac{5\pi}{6}$ is in QII, so reference angle = $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$

Step2: Reduce angle to coterminal in [0,2π)

$\frac{5\pi}{3}$ is already in [0,2π) and QIV, so reference angle = $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$

Step3: Adjust negative angle to positive coterminal

Add $2\pi$ to $-\frac{5\pi}{4}$: $-\frac{5\pi}{4} + 2\pi = -\frac{5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}$
$\frac{3\pi}{4}$ is in QII, so reference angle = $\pi - \frac{3\pi}{4} = \frac{\pi}{4}$

Answer:

$\frac{\pi}{6}$
$\frac{\pi}{3}$
$\frac{\pi}{4}$