Trigonometry Examples

$\cos (\frac{7}{8} \cdot π)$

Step 1

Combine $\frac{7}{8}$ and $π$ .

$\cos (\frac{7 π}{8})$

Step 2

The exact value of $\cos (\frac{7 π}{8})$ is $- \frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 2.1

Rewrite $\frac{7 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{\frac{7 π}{4}}{2})$

Step 2.2

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

$\pm \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}}$

Step 2.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}}$

Step 2.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{7 π}{4})}{2}}$ .

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Step 2.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Combine the numerators over the common denominator.

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

Step 2.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.6

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

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Step 2.4.6.1

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

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

Step 2.4.6.2

Multiply $2$ by $2$ .

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

Step 2.4.7

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

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

Step 2.4.8

Simplify the denominator.

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Step 2.4.8.1

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

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

Step 2.4.8.2

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.92387953 \dots$