Question

Find the value of the following trigonometric ratio:

\[\cos\left( - \frac{25\pi}{4} \right)\]

Answer 1

We have: 
\[\cos\left( - \frac{25\pi}{4} \right) = \cos\left( 1125^\circ \right)\]
\[\cos \left( - 1125^\circ \right) = \cos \left( 1125^\circ \right) = \cos \left( 90^\circ \times 12 + 45^\circ \right)\]
\[1125^\circ \text{ lies in the first quadrant in which the cosine function is positive . }\]
Also, 12 is an even integer .
\[ \therefore \cos\left( - 1125^\circ \right) = \cos\left( 1125^\circ \right) = \cos\left( 90^\circ \times 12 + 45^\circ \right) = \cos 45^\circ = \frac{1}{\sqrt{2}}\]