Questions: An angle, θ, has a reference angle of π/6 radians, and the angle θ is between π and 3π/2. Determine the exact angle measure of θ in radians.

Transcript text: 5. [10 pts] An angle, $\theta$, has a reference angle of $\frac{\pi}{6}$ radians, and the angle $\theta$ is between $\pi$ and $\frac{3 \pi}{2}$. Determine the exact angle measure of $\theta$ in radians.

Solution

Solution Steps

Step 1: Understand the reference angle

The reference angle is given as $\frac{\pi}{6}$ radians. A reference angle is the smallest angle between the terminal side of $\theta$ and the x-axis.

Step 2: Determine the quadrant of $\theta$

The problem states that $\theta$ lies between $\pi$ and $\frac{3\pi}{2}$, which is the third quadrant. In the third quadrant, the angle $\theta$ can be expressed as:
\[ \theta = \pi + \text{reference angle} = \pi + \frac{\pi}{6}. \]

Step 3: Calculate the exact angle measure

Substitute the reference angle into the expression:
\[ \theta = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}. \]
Thus, the exact measure of $\theta$ is $\frac{7\pi}{6}$ radians.

Final Answer

$ \boxed{\frac{7\pi}{6}} $

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