Trigonometry Examples

sin (\frac{π}{8})

$\sin (\frac{π}{8})$

Step 1

Rewrite

\frac{π}{8}

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

2

$2$ .

sin (\frac{\frac{π}{4}}{2})

$\sin (\frac{\frac{π}{4}}{2})$

Step 2

Apply the sine half-angle identity.

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

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

Step 3

Change the

\pm

$\pm$ to

+

$+$ because sine is positive in the first quadrant.

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

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

Step 4

Simplify

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

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

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Step 4.1

The exact value of

cos (\frac{π}{4})

$\cos (\frac{π}{4})$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

Step 4.2

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 4.3

Combine the numerators over the common denominator.

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

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

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 4.5

Multiply

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

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

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Step 4.5.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 4.5.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 4.6

Rewrite

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

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

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

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

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

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

Step 4.7

Simplify the denominator.

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Step 4.7.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 4.7.2

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

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

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

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.38268343 \dots

$0.38268343 \dots$