On the unit circle, a terminal ray starts at the positive x-axis and rotates counterclockw… — ACT

On the unit circle, a terminal ray starts at the positive $x$ -axis and rotates counterclockwise by $120^\circ$ , and then rotates counterclockwise by an additional $135^\circ$ . What is the radian measure of the final angle in standard position?

A
$\frac{5\pi}{12}$
B
$\frac{5\pi}{7}$
$\frac{17\pi}{12}$ Answer
D
$\frac{17\pi}{24}$
E
$\frac{45900}{\pi}$

Answer

\frac{17\pi}{12}

The correct option is

\frac{17\pi}{12}

. First, the two counterclockwise rotations are added to find the total angle in standard position:

120^\circ + 135^\circ = 255^\circ

. To convert this angle into radians, it is multiplied by the conversion ratio

\frac{\pi}{180^\circ}

, giving

\frac{255\pi}{180}

. Dividing the numerator and denominator by 15 simplifies the fraction to

\frac{17\pi}{12}

Step-by-Step Solution

Calculate the total counterclockwise rotation angle in degrees by summing the two individual angles.

120^\circ + 135^\circ = 255^\circ

Consecutive rotations in the same direction add together to find the final angle measure.

Convert the total angle from degrees to radians by multiplying by the conversion factor.

255^\circ \times \frac{\pi}{180^\circ} = \frac{255\pi}{180}

Since 180 degrees equals pi radians, multiplying by pi over 180 converts degrees to radians.

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

\frac{255\pi}{180} = \frac{17\pi}{12}

Simplifying the fraction by dividing both 255 and 180 by 15 yields the simplest form.

Key Concept

Unit Circle and Angle Measures (Radians and Degrees)

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