Trigonometry Examples

csc (60)

$\csc (60)$

Step 1

To convert degrees to radians, multiply by

$\frac{π}{180 °}$ , since a full circle is

$360 °$ or

$2 π$ radians.

Step 2

The exact value of

$\csc (60)$ is

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

$\frac{2}{\sqrt{3}} \cdot \frac{π}{180}$ radians

Step 3

Cancel the common factor of

$2$ .

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Step 3.1

Factor

$2$ out of

$180$ .

$\frac{2}{\sqrt{3}} \cdot \frac{π}{2 (90)}$ radians

Step 3.2

Cancel the common factor.

$\frac{2}{\sqrt{3}} \cdot \frac{π}{2 \cdot 90}$ radians

Step 3.3

Rewrite the expression.

$\frac{1}{\sqrt{3}} \cdot \frac{π}{90}$ radians

Step 4

Multiply

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

$\frac{π}{90}$ .

$\frac{π}{\sqrt{3} \cdot 90}$ radians

Step 5

Move

$90$ to the left of

$\sqrt{3}$ .

$\frac{π}{90 \sqrt{3}}$ radians

Step 6

Multiply

$\frac{π}{90 \sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{π}{90 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$ radians

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply

$\frac{π}{90 \sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{π \sqrt{3}}{90 \sqrt{3} \sqrt{3}}$ radians

Step 7.2

Move

$\sqrt{3}$ .

$\frac{π \sqrt{3}}{90 (\sqrt{3} \sqrt{3})}$ radians

Step 7.3

Raise

$\sqrt{3}$ to the power of

$1$ .

$\frac{π \sqrt{3}}{90 (\sqrt{3} \sqrt{3})}$ radians

Step 7.4

Raise

$\sqrt{3}$ to the power of

$1$ .

$\frac{π \sqrt{3}}{90 (\sqrt{3} \sqrt{3})}$ radians

Step 7.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{π \sqrt{3}}{90 {\sqrt{3}}^{1 + 1}}$ radians

Step 7.6

Add

$1$ and

$1$ .

$\frac{π \sqrt{3}}{90 {\sqrt{3}}^{2}}$ radians

Step 7.7

Rewrite

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

$3$ .

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Step 7.7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

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

$\frac{π \sqrt{3}}{90 {(3^{\frac{1}{2}})}^{2}}$ radians

Step 7.7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{π \sqrt{3}}{90 \cdot 3^{\frac{1}{2} \cdot 2}}$ radians

Step 7.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{π \sqrt{3}}{90 \cdot 3^{\frac{2}{2}}}$ radians

Step 7.7.4

Cancel the common factor of

$2$ .

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Step 7.7.4.1

Cancel the common factor.

$\frac{π \sqrt{3}}{90 \cdot 3^{\frac{2}{2}}}$ radians

Step 7.7.4.2

Rewrite the expression.

$\frac{π \sqrt{3}}{90 \cdot 3}$ radians

Step 7.7.5

Evaluate the exponent.

$\frac{π \sqrt{3}}{90 \cdot 3}$ radians

Step 8

Multiply

$90$ by

$3$ .

$\frac{π \sqrt{3}}{270}$ radians