Find the Exact Value cos(157.5 degrees )

$\cos (157.5 °)$

Rewrite $157.5 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{315}{2})$

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\pm \sqrt{\frac{1 + \cos (315)}{2}}$

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$- \sqrt{\frac{1 + \cos (315)}{2}}$

Simplify $- \sqrt{\frac{1 + \cos (315)}{2}}$ .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \sqrt{\frac{1 + \cos (45)}{2}}$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}}$

Multiply $\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{2 + \sqrt{2}}{2}$ by $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

$- \sqrt{\frac{2 + \sqrt{2}}{4}}$

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.92387953 \dots$