Precalculus Examples

$2 \cos (\frac{π}{8}) - 1$

Step 1

The exact value of $\cos (\frac{π}{8})$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.1

Rewrite $\frac{π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$2 \cos (\frac{\frac{π}{4}}{2}) - 1$

Step 1.2

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

$2 (\pm \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}}) - 1$

Step 1.3

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

$2 \sqrt{\frac{1 + \cos (\frac{π}{4})}{2}} - 1$

Step 1.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Combine the numerators over the common denominator.

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

Step 1.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.4

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

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Step 1.5.4.1

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

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

Step 1.5.4.2

Multiply $2$ by $2$ .

$2 \sqrt{\frac{2 + \sqrt{2}}{4}} - 1$

Step 1.5.5

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

$2 \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}} - 1$

Step 1.5.6

Simplify the denominator.

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Step 1.5.6.1

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

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

Step 1.5.6.2

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

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

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.84775906 \dots$