On the unit circle, what is the radian measure of the arc that subtends a central angle of $90°$?

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer: (C)

Verified step by step guidance

1. Recall that the radian measure of an angle is defined as the length of the arc on the unit circle subtended by that angle.
2. Understand the relationship between degrees and radians: \$180^\circ\( is equivalent to \)\pi$ radians.
3. Set up a proportion to convert the given angle from degrees to radians: if \$180^\circ = \pi\( radians, then \)90^\circ = x\( radians, so \)x = \frac{90}{180} \times \pi$.
4. Simplify the fraction \(\frac{90}{180}\) to \(\frac{1}{2}\), so the radian measure is \(\frac{1}{2} \pi\) or \(\frac{\pi}{2}\).
5. Conclude that the radian measure of the arc subtending a \$90^\circ\( central angle on the unit circle is \)\frac{\pi}{2}$.