Trigonometry Examples

$6713 \cos (210)$

Step 1

The exact value of $\cos (210)$ is $- \frac{\sqrt{3}}{2}$ .

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Step 1.1

Rewrite $210$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$6713 \cos (\frac{420}{2})$

Step 1.2

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

$6713 (\pm \sqrt{\frac{1 + \cos (420)}{2}})$

Step 1.3

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

$6713 (- \sqrt{\frac{1 + \cos (420)}{2}})$

Step 1.4

Simplify $- \sqrt{\frac{1 + \cos (420)}{2}}$ .

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Step 1.4.1

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

$6713 (- \sqrt{\frac{1 + \cos (60)}{2}})$

Step 1.4.2

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

$6713 (- \sqrt{\frac{1 + \frac{1}{2}}{2}})$

Step 1.4.3

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

$6713 (- \sqrt{\frac{\frac{2}{2} + \frac{1}{2}}{2}})$

Step 1.4.4

Combine the numerators over the common denominator.

$6713 (- \sqrt{\frac{\frac{2 + 1}{2}}{2}})$

Step 1.4.5

Add $2$ and $1$ .

$6713 (- \sqrt{\frac{\frac{3}{2}}{2}})$

Step 1.4.6

Multiply the numerator by the reciprocal of the denominator.

$6713 (- \sqrt{\frac{3}{2} \cdot \frac{1}{2}})$

Step 1.4.7

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

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Step 1.4.7.1

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

$6713 (- \sqrt{\frac{3}{2 \cdot 2}})$

Step 1.4.7.2

Multiply $2$ by $2$ .

$6713 (- \sqrt{\frac{3}{4}})$

Step 1.4.8

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

$6713 (- \frac{\sqrt{3}}{\sqrt{4}})$

Step 1.4.9

Simplify the denominator.

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Step 1.4.9.1

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

$6713 (- \frac{\sqrt{3}}{\sqrt{2^{2}}})$

Step 1.4.9.2

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

$6713 (- \frac{\sqrt{3}}{2})$

Step 2

Multiply $6713 (- \frac{\sqrt{3}}{2})$ .

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Step 2.1

Multiply $- 1$ by $6713$ .

$- 6713 \frac{\sqrt{3}}{2}$

Step 2.2

Combine $- 6713$ and $\frac{\sqrt{3}}{2}$ .

$\frac{- 6713 \sqrt{3}}{2}$

Step 3

Move the negative in front of the fraction.

$- \frac{6713 \sqrt{3}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{6713 \sqrt{3}}{2}$

Decimal Form:

$- 5813.62853560 \dots$