Question

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

Answer 1

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 \(\displaystyle\frac{{1}}{\sqrt{{2}}}\)