Task: Convert the angle $\alpha=510^{\circ}$ from degrees to radians.
Richard solved the problem in the following steps:
(1) Richard claimed that in the unit circle, the measure of a straight angle, i.e., $180^{\circ}$, corresponds to an arc length of $\pi$ radians.
(2) He expressed the angle $\alpha=510^{\circ}$ as the sum of multiples of the straight angle and the remining angle: $$\alpha=510^{\circ}=2\cdot 180^{\circ}+150^{\circ}$$
(3) In this decomposition, he expressed $180^{\circ}$ as $\pi$ radians and $150^{\circ}$ as $\frac16$ of a straight angle: $$\alpha=2\pi+\frac{\pi}{6}$$
(4) Finally, he summed up the individually converted parts of the angle and obtained: $$\alpha=\frac{13}{6}\pi$$ Did Richard solve the task correctly? If not, identify the step in which he made a mistake.
Yes. The whole solution is correct.
No. The mistake is in step (1). A different arc length corresponds to the straight angle in the unit circle.
No. The mistake is in step (2). The remining angle should have been expressed in terms of the measure of the corresponding acute angle.
No. The mistake is in step (3). The conversion of degrees to radians is not done correctly in this step.
No. The mistake is in step (4). The addition is incorrectly performed.
The correct procedure is: \begin{aligned} &\alpha=510^{\circ}=2\cdot180^{\circ}+150^{\circ}\cr &α=2\pi+\frac{5\pi}{6}=\frac{17}{6}\pi \end{aligned}
Alternative solution:
We know that $180$ degrees is equivalent to $\pi$ radians. Therefore, $1$ degree is equivalent to $\frac{\pi}{180}$ radians. To convert $510$ degrees to radians, we multiply $510$ by $\frac{\pi}{180}$: $$\alpha=510^{\circ}\Rightarrow\alpha=510\cdot\frac{\pi}{180}\mbox{radians}=\frac{17\pi}{6}\mbox{radians}$$