Find the Exact Value cos(22.5 degrees )

$\cos (22.5 °)$

Step 1

Rewrite $22.5 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{45}{2})$

Step 2

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

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

Step 3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

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

Step 4

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

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

Step 5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

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.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.92387953 \dots$