Precalculus Examples

$\sin (- 285)$

Step 1

Rewrite $- 285$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sin (\frac{- 570}{2})$

Step 2

Apply the sine half-angle identity.

$\pm \sqrt{\frac{1 - \cos (- 570)}{2}}$

Step 3

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

$\sqrt{\frac{1 - \cos (- 570)}{2}}$

Step 4

Simplify $\sqrt{\frac{1 - \cos (- 570)}{2}}$ .

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Step 4.1

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sqrt{\frac{1 - \cos (150)}{2}}$

Step 4.2

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.

$\sqrt{\frac{1 - - \cos (30)}{2}}$

Step 4.3

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{2}}$

Step 4.4

Multiply $- - \frac{\sqrt{3}}{2}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{2}}$

Step 4.4.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}$

Step 4.5

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

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}}$

Step 4.6

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}}$

Step 4.7

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}}$

Step 4.8

Multiply $\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}$ .

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Step 4.8.1

Multiply $\frac{2 + \sqrt{3}}{2}$ by $\frac{1}{2}$ .

$\sqrt{\frac{2 + \sqrt{3}}{2 \cdot 2}}$

Step 4.8.2

Multiply $2$ by $2$ .

$\sqrt{\frac{2 + \sqrt{3}}{4}}$

Step 4.9

Rewrite $\sqrt{\frac{2 + \sqrt{3}}{4}}$ as $\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}$ .

$\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}$

Step 4.10

Simplify the denominator.

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Step 4.10.1

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

$\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{2^{2}}}$

Step 4.10.2

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

$\frac{\sqrt{2 + \sqrt{3}}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2 + \sqrt{3}}}{2}$

Decimal Form:

$0.96592582 \dots$