Question

convert the angle $\theta = \dfrac{257\pi}{360}$ radians to degrees. express your answer exactly. $\theta = \square^\circ$

Explanation:

Step1: Recall radian to degree conversion formula

To convert radians to degrees, we use the formula: \( \text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi} \).

Step2: Substitute the given radian value

We have \( \theta = \frac{257\pi}{360} \) radians. Substitute this into the formula:
\( \theta = \frac{257\pi}{360} \times \frac{180^\circ}{\pi} \)

Step3: Simplify the expression

The \( \pi \) terms cancel out, and \( \frac{180}{360} = \frac{1}{2} \). So we have:
\( \theta = 257 \times \frac{180^\circ}{360} = 257 \times \frac{1}{2}^\circ = \frac{257}{2}^\circ = 128.5^\circ \)? Wait, no, wait. Wait, \( \frac{180}{360} = \frac{1}{2} \)? Wait, no, \( 180 \div 360 = 0.5 \), but let's do it again. Wait, \( \frac{257\pi}{360} \times \frac{180}{\pi} \). The \( \pi \) cancels, then \( 180/360 = 1/2 \), so \( 257 \times (180/360) = 257 \times (1/2) \)? Wait, no, 180 divided by 360 is 1/2, so 257 times 1/2 is 128.5? Wait, no, that can't be. Wait, no, wait, 180/360 is 1/2, but let's check the formula again. The conversion factor is \( \frac{180^\circ}{\pi} \) radians⁻¹. So when you multiply radians by \( \frac{180^\circ}{\pi} \), you get degrees. So \( \frac{257\pi}{360} \times \frac{180^\circ}{\pi} \). The \( \pi \) cancels, 180 and 360 simplify: 180/360 = 1/2, so 257 (180/360) = 257 (1/2) = 128.5? Wait, no, 257 times 180 is 46260, divided by 360 is 46260/360 = 128.5? Wait, 360*128 = 46080, 46260 - 46080 = 180, so 128 + 180/360 = 128.5. So \( \theta = 128.5^\circ \)? Wait, but let's do it again. \( \frac{257\pi}{360} \times \frac{180}{\pi} = \frac{257 \times 180}{360} = \frac{257}{2} = 128.5 \). So the angle in degrees is \( \frac{257}{2} \) degrees or 128.5 degrees.

Answer:

\( \frac{257}{2} \) (or 128.5)