Question

homework assignment #7.4
score: 2/20 2/20 answered
question 3
find the exact value of each of the following using the reference angle theorem rather than a calculator.
(i) if \\( \theta = \frac{5\pi}{3} \\) then \\( \hat{\theta} = \\)
(ii) if \\( \theta = \frac{7\pi}{6} \\) then \\( \hat{\theta} = \\)
(iii) if \\( \theta = \frac{-5\pi}{4} \\) then \\( \hat{\theta} = \\)

Explanation:

Step1: Determine quadrant of $\theta=\frac{5\pi}{3}$

$\frac{5\pi}{3}$ is in Quadrant IV (since $\frac{3\pi}{2}<\frac{5\pi}{3}<2\pi$). Reference angle formula for Quadrant IV: $\theta' = 2\pi - \theta$.
$\theta' = 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}$

Step2: Determine quadrant of $\theta=\frac{7\pi}{6}$

$\frac{7\pi}{6}$ is in Quadrant III (since $\pi<\frac{7\pi}{6}<\frac{3\pi}{2}$). Reference angle formula for Quadrant III: $\theta' = \theta - \pi$.
$\theta' = \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}$

Step3: Determine quadrant of $\theta=-\frac{5\pi}{4}$

Add $2\pi$ to make it positive: $-\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4}$, which is in Quadrant II. Reference angle formula for Quadrant II: $\theta' = \pi - \theta$.
$\theta' = \pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4}$

Answer:

(i) $\frac{\pi}{3}$
(ii) $\frac{\pi}{6}$
(iii) $\frac{\pi}{4}$