Trigonometry Examples

$\cos (- 330 °)$

Step 1

Rewrite $- 330 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{- 660}{2})$

Step 2

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

$\pm \sqrt{\frac{1 + \cos (- 660)}{2}}$

Step 3

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

$\sqrt{\frac{1 + \cos (- 660)}{2}}$

Step 4

Simplify $\sqrt{\frac{1 + \cos (- 660)}{2}}$ .

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Step 4.1

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

$\sqrt{\frac{1 + \cos (60)}{2}}$

Step 4.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Add $2$ and $1$ .

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

Step 4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Multiply $2$ by $2$ .

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

Step 4.8

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

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

Step 4.9

Simplify the denominator.

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Step 4.9.1

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

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

Step 4.9.2

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$