Trigonometry Examples

$\sqrt{\frac{- \frac{3}{5}}{1 + \frac{4}{5}}}$

Step 1

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

$\sqrt{\frac{- \frac{3}{5}}{\frac{5}{5} + \frac{4}{5}}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{- \frac{3}{5}}{\frac{5 + 4}{5}}}$

Step 3

Add $5$ and $4$ .

$\sqrt{\frac{- \frac{3}{5}}{\frac{9}{5}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{- \frac{3}{5} \cdot \frac{5}{9}}$

Step 5

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$\sqrt{\frac{- 3}{5} \cdot \frac{5}{9}}$

Step 5.2

Factor $3$ out of $- 3$ .

$\sqrt{\frac{3 (- 1)}{5} \cdot \frac{5}{9}}$

Step 5.3

Factor $3$ out of $9$ .

$\sqrt{\frac{3 \cdot - 1}{5} \cdot \frac{5}{3 \cdot 3}}$

Step 5.4

Cancel the common factor.

$\sqrt{\frac{3 \cdot - 1}{5} \cdot \frac{5}{3 \cdot 3}}$

Step 5.5

Rewrite the expression.

$\sqrt{\frac{- 1}{5} \cdot \frac{5}{3}}$

Step 6

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\sqrt{\frac{- 1}{5} \cdot \frac{5}{3}}$

Step 6.2

Rewrite the expression.

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

Step 7

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

Tap for more steps...

Step 7.1

Rewrite $- 1$ as $i^{2}$ .

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

Step 7.2

Rewrite $1$ as $1^{2}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

One to any power is one.

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

Step 10

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 11

Any root of $1$ is $1$ .

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

Step 12

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

$i (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 13

Combine and simplify the denominator.

Tap for more steps...

Step 13.1

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

$i \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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

Step 13.5

Add $1$ and $1$ .

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

Step 13.6

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

Tap for more steps...

Step 13.6.1

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

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

Step 13.6.2

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

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

Step 13.6.3

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

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

Step 13.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.6.4.1

Cancel the common factor.

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

Step 13.6.4.2

Rewrite the expression.

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

Step 13.6.5

Evaluate the exponent.

$i \frac{\sqrt{3}}{3}$

Step 14

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$\frac{i \sqrt{3}}{3}$