Question

Find the value of the following trigonometric ratio :

\[ \cos\left(\frac{19\pi}{4}\right) \]

Solution

\[ \cos\left(\frac{19\pi}{4}\right) = \cos\left(\frac{19\pi}{4}-4\pi\right) \]

\[ = \cos\frac{3\pi}{4} \]

\[ = \cos\left(\pi-\frac{\pi}{4}\right) \]

Using,

\[ \cos(\pi-\theta)=-\cos\theta \]

\[ = -\cos\frac{\pi}{4} = -\frac{1}{\sqrt2} = -\frac{\sqrt2}{2} \]

Answer :

\[ \boxed{ \cos\left(\frac{19\pi}{4}\right) = -\frac{\sqrt2}{2} } \]