Trigonometry Examples

$\sqrt{\frac{1 - \cos (\frac{7 π}{8})}{2}}$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Combine the numerators over the common denominator.

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

Step 1.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

Multiply $2$ by $2$ .

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

Step 1.4.7

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

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

Step 1.4.8

Simplify the denominator.

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Step 1.4.8.1

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

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

Step 1.4.8.2

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

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

Step 2

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

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Step 2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $2$ by $2$ .

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

Step 7

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

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

Step 8

Simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.98078528 \dots$