Trigonometry Examples

$\cos (22.5)$

Step 1

The supplement of $\cos (22.5)$ is the angle that when added to $\cos (22.5)$ forms a straight angle ( $180 °$ ).

$180 ° - \cos (22.5)$

Step 2

The exact value of $\cos (22.5)$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 2.1

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

Step 2.2

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

Step 2.3

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

Step 2.4

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

Step 2.5

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

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Step 2.5.1

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

Step 2.5.2

Combine the numerators over the common denominator.

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

Step 2.5.4

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

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Step 2.5.4.1

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

Step 2.5.4.2

Multiply $2$ by $2$ .

Step 2.5.5

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

Step 2.5.6

Simplify the denominator.

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Step 2.5.6.1

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

Step 2.5.6.2

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

$180 - \frac{\sqrt{2 + \sqrt{2}}}{2 °}$

Step 3

To write $180$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$180 \cdot \frac{2}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2 °}$

Step 4

Combine $180$ and $\frac{2}{2}$ .

$\frac{180 \cdot 2}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2 °}$

Step 5

Simplify the expression.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{180 \cdot 2 - \sqrt{2 + \sqrt{2}}}{2 °}$

Step 5.2

Multiply $180$ by $2$ .

$\frac{360 - \sqrt{2 + \sqrt{2}}}{2 °}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{360 - \sqrt{2 + \sqrt{2}}}{2 °}$

Decimal Form:

$179.07612046 \dots$