Question

Find the exact value of the trigonometric function cos (9pi)/4.

Trigonometric Functions
Find the exact value of the trigonometric function $$\displaystyle\frac{{\cos{{\left({9}\pi\right)}}}}{{4}}$$.

2021-01-28
Rewriting the given trigonometric function,
$$\displaystyle\frac{{\cos{{\left({9}\pi\right)}}}}{{4}}={\cos{{\left(\frac{{{8}\pi+\pi}}{{4}}\right)}}}$$
$$\displaystyle={\cos{{\left(\frac{{{8}\pi}}{{4}}+\frac{\pi}{{4}}\right)}}}$$
$$\displaystyle={\cos{{\left({2}\pi+\frac{\pi}{{4}}\right)}}}$$
We know that $$\displaystyle{\left({2}\pi+\theta\right)}$$ always lies in first quadrant and in first quadrant all trigonometric functions are positive.
So applying above rule for the given trigonometric function, we get
$$\displaystyle{\cos{{\left({2}\pi+\frac{\pi}{{4}}\right)}}}={\cos{{\left(\frac{\pi}{{4}}\right)}}}$$
$$\displaystyle=\frac{{1}}{\sqrt{{2}}}$$
Hence, exact value of the given trigonometric function is $$\displaystyle\frac{{1}}{\sqrt{{2}}}$$