An angle in standard position measures \theta = \frac{7\pi}{6} radians. If the terminal si… — ACT

An angle in standard position measures $\theta = \frac{7\pi}{6}$ radians. If the terminal side of the angle is rotated counterclockwise by $120^\circ$ , what is the radian measure of the final angle?

A
$\frac{3\pi}{2}$
B
$\pi$
C
$\frac{\pi}{2}$
$\frac{11\pi}{6}$ Answer
E
$\frac{8\pi}{3}$

Answer

The final angle measures

\frac{11\pi}{6}

radians.

The correct answer is

\frac{11\pi}{6}

radians. To find the final angle, we convert the rotation angle of

120^\circ

into radians:

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians. Since the rotation is counterclockwise, we add this value to the initial angle of

\frac{7\pi}{6}

radians:

\frac{7\pi}{6} + \frac{2\pi}{3} = \frac{7\pi}{6} + \frac{4\pi}{6} = \frac{11\pi}{6}

radians.

Step-by-Step Solution

Convert the rotation angle from degrees to radians.

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians.

Both angles must be in radians to add them directly.

Set up the sum for a counterclockwise rotation.

\frac{7\pi}{6} + \frac{2\pi}{3}

Counterclockwise rotation corresponds to adding the angle measure.

Find the common denominator and calculate the sum.

\frac{7\pi}{6} + \frac{4\pi}{6} = \frac{11\pi}{6}

radians.

The least common denominator of

6

and

3

6

, so we multiply the numerator and denominator of

\frac{2\pi}{3}

2

before adding.

Key Concept

Unit Circle and Angle Measures (Radians and Degrees)

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic