Precalculus Examples

$\frac{π}{8}$

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 $2$ .

$\cos (\frac{\frac{π}{4}}{2})$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

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.

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

Step 1.5.2

Combine the numerators over the common denominator.

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

Step 1.5.3

Multiply the numerator by the reciprocal of the denominator.

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

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 $\frac{1}{2}$ .

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

Step 1.5.4.2

Multiply $2$ by $2$ .

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

Step 1.5.5

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

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

Step 1.5.6

Simplify the denominator.

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Step 1.5.6.1

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

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

Step 1.5.6.2

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

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

Step 2

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

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Step 2.1

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

$\sin (\frac{\frac{π}{4}}{2})$

Step 2.2

Apply the sine half-angle identity.

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Combine the numerators over the common denominator.

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

Step 2.4.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

Multiply $2$ by $2$ .

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

Step 2.4.6

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

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

Step 2.4.7

Simplify the denominator.

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Step 2.4.7.1

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

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

Step 2.4.7.2

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

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

Step 3

Set up the coordinates $(\cos (θ), \sin (θ))$ .

$(\frac{\sqrt{2 + \sqrt{2}}}{2}, \frac{\sqrt{2 - \sqrt{2}}}{2})$