Question

Assume that $\theta$ is rational multiple of $\pi$ such that $\cos \theta$ is a distinct rational. The number of values of $\cos \theta$ is

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (C)

$5$

In the range of values of $cos\theta$ the only rational numbers are

$\{-1,\frac{-4}{5},\frac{-3}{5},-\frac{-1}{2},0,1,\frac{1}{2},\frac{3}{5},\frac{4}{5},1\}$

However

$co{s}^{-1}\left(\frac{\pm 3}{5}\right)$ and $co{s}^{-1}\left(\frac{\pm 4}{5}\right)$ are not ratioanl multiples of $\pi$

Therefore the modified range is

$\{-1,\frac{-1}{2},0,\frac{1}{2},1\}$

Hence there are in total 5 values.