An angle in standard position measures \frac{2\pi}{3} radians. If the angle is increased b… — ACT

An angle in standard position measures $\frac{2\pi}{3}$ radians. If the angle is increased by $45^\circ$ , what is the measure of the new angle, in radians?

A
$\frac{3\pi}{7}$
B
$\frac{14\pi}{3}$
C
$\frac{12\pi}{11}$
$\frac{11\pi}{12}$ Answer
E
$\frac{3\pi}{4}$

Answer

The correct answer is

\frac{11\pi}{12}

To find the final angle measure, first convert the rotation angle of

45^\circ

into radians. Since

180^\circ = \pi

radians, multiplying

45^\circ

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

gives

\frac{\pi}{4}

radians. Next, add the initial angle and the rotation:

\frac{2\pi}{3} + \frac{\pi}{4}

. Finding a common denominator of

12

, the sum is

\frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians.

Step-by-Step Solution

Convert the rotation angle from degrees to radians.

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians

To perform the addition, both angle measures must be in the same unit (radians).

Add the two radian measures.

\frac{2\pi}{3} + \frac{\pi}{4} = \frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians

An increase in angle measure corresponds to counterclockwise rotation, which means adding the two angles.

Key Concept

Converting between degrees and radians and adding angles in standard position.

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

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