If an angle in standard position intercepts an arc $\mathrm{PQ}$ on a circle of radius $1$ such that the length of arc $\mathrm{PQ}$ is $0.42$ units, what is the measure of the angle in degrees?

• A. $33°$
• B. $24°$
• C. $48°$
• D. $66°$

Answer: (A)

Verified step by step guidance

1. Recall the formula that relates the length of an arc (s), the radius of the circle (r), and the central angle in radians (θ): \(s = r \times \theta\).
2. Since the radius \(r\) is given as 1 unit, substitute \(r = 1\) and the arc length \(s = 0.42\) into the formula to find the angle in radians: \(0.42 = 1 \times \theta\), so \(\theta = 0.42\) radians.
3. Convert the angle from radians to degrees using the conversion factor \(180^\circ / \pi\): \(\theta_{degrees} = \theta_{radians} \times \frac{180^\circ}{\pi}\).
4. Substitute \(\theta = 0.42\) radians into the conversion formula: \(\theta_{degrees} = 0.42 \times \frac{180^\circ}{\pi}\).
5. Calculate the value from the previous step to find the angle in degrees, then compare it with the given options to select the closest match.