Trigonometry Examples

$\csc (- 330)$

Step 1

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

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

Step 2

Apply the reciprocal identity to $\csc (\frac{- 660}{2})$ .

$\frac{1}{\sin (\frac{- 660}{2})}$

Step 3

Apply the sine half-angle identity.

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

Step 4

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

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

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Subtract $1$ from $2$ .

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $2$ by $2$ .

$\frac{1}{\sqrt{\frac{1}{4}}}$

Step 5.2.3

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{1}{\frac{\sqrt{1}}{\sqrt{4}}}$

Step 5.2.4

Any root of $1$ is $1$ .

$\frac{1}{\frac{1}{\sqrt{4}}}$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

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

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

Step 5.2.5.2

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

$\frac{1}{\frac{1}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 5.4

Multiply $2$ by $1$ .

$2$