Question

Let $\theta$ be a positive angle. If the number of degrees in $\theta$ is divided by the number of radians in $\theta$, then an irrational number $180/\pi$ results. If the number of degrees in $\theta$ is multiplied by the number of radians in $\theta$, then an irrational number $125\pi /9$ results. The angle $\theta$ must be equal to.

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${50}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

Answer: (C)

${50}^{\circ }$

Let ${\theta }_d$ and ${\theta }_r$ be the angle in degrees and radians respectively.

$\therefore {\theta }_r=\frac{\pi }{180}{\theta }_d$ ... (1)

Also, we have ${\theta }_d\times {\theta }_r=\frac{125\pi }{9}$ ... (2)

from (1) and (2), we get

${\theta }_d^2\frac{\pi }{180}=\frac{125\pi }{9}⟹{\theta }_d^2=2500⟹{\theta }_d={50}^{\circ }$

This is the required solution.