Questions: Sketch the given angle θ in standard position and find its reference angle in degrees and radians. θ = 5π/6 The reference angle in radians is π/6. (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.) The reference angle in degrees is 30°. (Simplify your answer.)

$Sketch the given angle θ in standard position and find its reference angle in degrees and radians. θ = 5π/6 The reference angle in radians is π/6. (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.) The reference angle in degrees is 30°. (Simplify your answer.)$

Transcript text: Sketch the given angle $\theta$ in standard position and find its reference angle in degrees and radians. \[ \theta=\frac{5 \pi}{6} \] The reference angle in radians is $\frac{\pi}{6}$. (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.) The reference angle in degrees is $\square$ $7^{\circ}$. (Simplify your answer.)

Solution

Solution Steps

Step 1: Identify the given angle in radians

The given angle is $ \theta = \frac{5\pi}{6} $.

Step 2: Determine the standard position of the angle

To sketch the angle $ \theta = \frac{5\pi}{6} $ in standard position, note that it is in the second quadrant because $ \frac{5\pi}{6} $ is between $ \pi/2 $ and $ \pi $.

Step 3: Choose the correct graph

The correct graph is the one that shows the angle $ \frac{5\pi}{6} $ in the second quadrant. This corresponds to option D.

Step 4: Find the reference angle in radians

The reference angle $ \theta' $ is the acute angle formed with the x-axis. For an angle in the second quadrant, the reference angle is $ \pi - \theta $. \[ \theta' = \pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6} \]

Step 5: Convert the reference angle to degrees

To convert the reference angle from radians to degrees, use the conversion factor $ 180^\circ = \pi $ radians. \[ \theta' = \frac{\pi}{6} \times \frac{180^\circ}{\pi} = 30^\circ \]