Trigonometry Examples

$\csc (- 300)$

Step 1

Rewrite $- 300$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\csc (\frac{- 600}{2})$

Step 2

Apply the reciprocal identity to $\csc (\frac{- 600}{2})$ .

$\frac{1}{\sin (\frac{- 600}{2})}$

Step 3

Apply the sine half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 600)}{2}}}$

Step 4

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

$\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{2}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{2}}}$ .

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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 $360$ °.

$\frac{1}{\sqrt{\frac{1 - \cos (120)}{2}}}$

Step 5.1.2

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.

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

Step 5.1.3

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

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

Step 5.1.4

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

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Step 5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.4.2

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

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

Step 5.1.5

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

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

Step 5.1.6

Combine the numerators over the common denominator.

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

Step 5.1.7

Add $2$ and $1$ .

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $2$ by $2$ .

$\frac{1}{\sqrt{\frac{3}{4}}}$

Step 5.2.3

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

$\frac{1}{\frac{\sqrt{3}}{\sqrt{4}}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

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

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

Step 5.2.4.2

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

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

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

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

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

Step 5.6.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.6.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.6.5

Add $1$ and $1$ .

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

Step 5.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.6.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

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

Step 5.6.6.4.2

Rewrite the expression.

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

Step 5.6.6.5

Evaluate the exponent.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.15470053 \dots$