Trigonometry Examples

$\csc (420)$

Step 1

To convert degrees to radians, multiply by $\frac{π}{180 °}$ , since a full circle is $360 °$ or $2 π$ radians.

Step 2

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

$\csc (60) \cdot \frac{π}{180}$ radians

Step 3

The exact value of $\csc (60)$ is $\frac{2}{\sqrt{3}}$ .

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

Step 4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $180$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{π}{90}$ .

$\frac{π}{\sqrt{3} \cdot 90}$ radians

Step 6

Move $90$ to the left of $\sqrt{3}$ .

$\frac{π}{90 \sqrt{3}}$ radians

Step 7

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 8

Combine and simplify the denominator.

Tap for more steps...

Step 8.1

Multiply $\frac{π}{90 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 8.2

Move $\sqrt{3}$ .

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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 8.6

Add $1$ and $1$ .

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

Step 8.7

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

Tap for more steps...

Step 8.7.1

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 8.7.2

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 8.7.3

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

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

Step 8.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.7.4.1

Cancel the common factor.

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

Step 8.7.4.2

Rewrite the expression.

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

Step 8.7.5

Evaluate the exponent.

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

Step 9

Multiply $90$ by $3$ .

$\frac{π \sqrt{3}}{270}$ radians