Trigonometry Examples

$\sin (67.5)$

Step 1

Rewrite

$67.5$ as an angle where the values of the six trigonometric functions are known divided by

$2$ .

$\sin (\frac{135}{2})$

Step 2

Apply the sine half-angle identity.

$\pm \sqrt{\frac{1 - \cos (135)}{2}}$

Step 3

Change the

$\pm$ to

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

$\sqrt{\frac{1 - \cos (135)}{2}}$

Step 4

Simplify

$\sqrt{\frac{1 - \cos (135)}{2}}$ .

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Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\sqrt{\frac{1 - - \cos (45)}{2}}$

Step 4.2

The exact value of

$\cos (45)$ is

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

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

Step 4.3

Multiply

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

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Step 4.3.1

Multiply

$- 1$ by

$- 1$ .

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

Step 4.3.2

Multiply

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

$1$ .

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

Step 4.4

Write

$1$ as a fraction with a common denominator.

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.7

Multiply

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

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Step 4.7.1

Multiply

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

$\frac{1}{2}$ .

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

Step 4.7.2

Multiply

$2$ by

$2$ .

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

Step 4.8

Rewrite

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

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

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

Step 4.9

Simplify the denominator.

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Step 4.9.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 4.9.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.92387953 \dots$