Question

find the exact value of \\(\cos(-225^{\circ})\\). \\(\cos(-225^{\circ}) = \\)

Explanation:

Step1: Use cosine even identity

$\cos(-\theta) = \cos(\theta)$
$\cos(-225^\circ) = \cos(225^\circ)$

Step2: Rewrite angle as sum

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

Step3: Apply cosine addition formula

$\cos(A+B) = \cos A\cos B - \sin A\sin B$
$\cos(180^\circ+45^\circ) = \cos180^\circ\cos45^\circ - \sin180^\circ\sin45^\circ$

Step4: Substitute known values

$\cos180^\circ=-1$, $\cos45^\circ=\frac{\sqrt{2}}{2}$, $\sin180^\circ=0$, $\sin45^\circ=\frac{\sqrt{2}}{2}$
$\cos(180^\circ+45^\circ) = (-1)\cdot\frac{\sqrt{2}}{2} - 0\cdot\frac{\sqrt{2}}{2}$

Step5: Simplify the expression

$= -\frac{\sqrt{2}}{2} - 0 = -\frac{\sqrt{2}}{2}$

Answer:

$-\frac{\sqrt{2}}{2}$