Question

Find the value of cos ${225}^{\circ }$

Answer 1

Evaluating the value of $\cos {225}^{\circ }$

$\cos {225}^{\circ }$ can be expressed as

$\cos {225}^{\circ }=\cos ({180}^{\circ }+{45}^{\circ })$

$\Rightarrow$$\cos ({180}^{\circ }+{45}^{\circ })=\cos {180}^{\circ }\times \cos {45}^{\circ }–\sin {180}^{\circ }\times \sin {45}^{\circ }$ ; $\left[\cos (A+B)=\cos \left(A\right)\times \cos \left(B\right)-\sin \left(A\right)\times \sin \left(B\right)\right]$

Substituting the values of the above angles we get,

$\cos {180}^{\circ }=-1,\cos {45}^{\circ }=\frac{\sqrt{2}}{2},\sin {180}^{\circ }=0\mathrm{and}\sin {45}^{\circ }=\frac{\sqrt{2}}{2}$

$\begin{array}{rcl}\cos ({180}^{\circ }+{45}^{\circ })& =& -1\times \left(\frac{\sqrt{2}}{2}\right)-0\times \left(\frac{\sqrt{2}}{2}\right)\\ & =& -\frac{\sqrt{2}}{2}\\ & =& -\frac{1}{\sqrt{2}}\\ & & \end{array}$

Hence the value of $\cos {225}^{\circ }$ is $-\frac{1}{\sqrt{2}}$