Precalculus Examples

$\cos (67.5)$

Step 1

Rewrite $67.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{135}{2})$

Step 2

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

$\pm \sqrt{\frac{1 + \cos (135)}{2}}$

Step 3

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

$\sqrt{\frac{1 + \cos (135)}{2}}$

Step 4

Simplify $\sqrt{\frac{1 + \cos (135)}{2}}$ .

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Multiply $2$ by $2$ .

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

Step 4.7

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

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

Step 4.8

Simplify the denominator.

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Step 4.8.1

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

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

Step 4.8.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.38268343 \dots$