Precalculus Examples

$\frac{\tan (45) + \tan (15)}{1 - \tan (45) \tan (15)}$

Step 1

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

The exact value of $\tan (45)$ is $1$ .

$\frac{1 + \tan (15)}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 2.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{1 + \tan (45 - 30)}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.2

Separate negation.

$\frac{1 + \tan (45 - (30))}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.3

Apply the difference of angles identity.

$\frac{1 + \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.4

The exact value of $\tan (45)$ is $1$ .

$\frac{1 + \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{1 + \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.6

The exact value of $\tan (45)$ is $1$ .

$\frac{1 + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{1 + \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 (\frac{\sqrt{3}}{3})}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8

Simplify $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ .

Tap for more steps...

Step 2.2.8.1

Multiply the numerator and denominator of the fraction by $3$ .

Tap for more steps...

Step 2.2.8.1.1

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

$\frac{1 + \frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 (\frac{\sqrt{3}}{3})}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.1.2

Combine.

$\frac{1 + \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.2

Apply the distributive property.

$\frac{1 + \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.8.3.1

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

$\frac{1 + \frac{3 \cdot 1 + 3 (\frac{- \sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.3.2

Cancel the common factor.

$\frac{1 + \frac{3 \cdot 1 + 3 (\frac{- \sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.3.3

Rewrite the expression.

$\frac{1 + \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.4

Multiply $3$ by $1$ .

$\frac{1 + \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.5

Simplify the denominator.

Tap for more steps...

Step 2.2.8.5.1

Multiply $3$ by $1$ .

$\frac{1 + \frac{3 - \sqrt{3}}{3 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{1 + \frac{3 - \sqrt{3}}{3 + 3 (1) (\frac{\sqrt{3}}{3})}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.5.2.2

Cancel the common factor.

$\frac{1 + \frac{3 - \sqrt{3}}{3 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.5.2.3

Rewrite the expression.

$\frac{1 + \frac{3 - \sqrt{3}}{3 + \sqrt{3}}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.6

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{1 + \frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.7

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.8

Expand the denominator using the FOIL method.

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.9

Simplify.

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.10

Simplify the numerator.

Tap for more steps...

Step 2.2.8.10.1

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

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.10.2

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

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.10.3

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

$\frac{1 + \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.10.4

Add $1$ and $1$ .

$\frac{1 + \frac{{(3 - \sqrt{3})}^{2}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.11

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

$\frac{1 + \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.12

Expand $(3 - \sqrt{3}) (3 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 2.2.8.12.1

Apply the distributive property.

$\frac{1 + \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.12.2

Apply the distributive property.

$\frac{1 + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.12.3

Apply the distributive property.

$\frac{1 + \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13

Simplify and combine like terms.

Tap for more steps...

Step 2.2.8.13.1

Simplify each term.

Tap for more steps...

Step 2.2.8.13.1.1

Multiply $3$ by $3$ .

$\frac{1 + \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{1 + \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 2.2.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4.2

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4.3

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4.4

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4.5

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.4.6

Add $1$ and $1$ .

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5

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

Tap for more steps...

Step 2.2.8.13.1.5.1

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5.2

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5.3

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

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.8.13.1.5.4.1

Cancel the common factor.

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5.4.2

Rewrite the expression.

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.1.5.5

Evaluate the exponent.

$\frac{1 + \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.2

Add $9$ and $3$ .

$\frac{1 + \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{1 + \frac{12 - 6 \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14

Cancel the common factor of $12 - 6 \sqrt{3}$ and $6$ .

Tap for more steps...

Step 2.2.8.14.1

Factor $6$ out of $12$ .

$\frac{1 + \frac{6 \cdot 2 - 6 \sqrt{3}}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.2

Factor $6$ out of $- 6 \sqrt{3}$ .

$\frac{1 + \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.3

Factor $6$ out of $6 (2) + 6 (- \sqrt{3})$ .

$\frac{1 + \frac{6 (2 - \sqrt{3})}{6}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.4

Cancel the common factors.

Tap for more steps...

Step 2.2.8.14.4.1

Factor $6$ out of $6$ .

$\frac{1 + \frac{6 (2 - \sqrt{3})}{6 (1)}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.4.2

Cancel the common factor.

$\frac{1 + \frac{6 (2 - \sqrt{3})}{6 \cdot 1}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.4.3

Rewrite the expression.

$\frac{1 + \frac{2 - \sqrt{3}}{1}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.2.8.14.4.4

Divide $2 - \sqrt{3}$ by $1$ .

$\frac{1 + 2 - \sqrt{3}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 2.3

Add $1$ and $2$ .

$\frac{3 - \sqrt{3}}{1 - \tan (45) \tan (15)} \cdot \frac{π}{180}$ radians

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

The exact value of $\tan (45)$ is $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \cdot (1 \tan (15))} \cdot \frac{π}{180}$ radians

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \tan (15)} \cdot \frac{π}{180}$ radians

Step 3.3

The exact value of $\tan (15)$ is $2 - \sqrt{3}$ .

Tap for more steps...

Step 3.3.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{3 - \sqrt{3}}{1 - 1 \tan (45 - 30)} \cdot \frac{π}{180}$ radians

Step 3.3.2

Separate negation.

$\frac{3 - \sqrt{3}}{1 - 1 \tan (45 - (30))} \cdot \frac{π}{180}$ radians

Step 3.3.3

Apply the difference of angles identity.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{\tan (45) - \tan (30)}{1 + \tan (45) \tan (30)}} \cdot \frac{π}{180}$ radians

Step 3.3.4

The exact value of $\tan (45)$ is $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{1 - \tan (30)}{1 + \tan (45) \tan (30)}} \cdot \frac{π}{180}$ radians

Step 3.3.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (45) \tan (30)}} \cdot \frac{π}{180}$ radians

Step 3.3.6

The exact value of $\tan (45)$ is $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (30)}} \cdot \frac{π}{180}$ radians

Step 3.3.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 (\frac{\sqrt{3}}{3})}} \cdot \frac{π}{180}$ radians

Step 3.3.8

Simplify $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \frac{\sqrt{3}}{3}}$ .

Tap for more steps...

Step 3.3.8.1

Multiply the numerator and denominator of the fraction by $3$ .

Tap for more steps...

Step 3.3.8.1.1

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

$\frac{3 - \sqrt{3}}{1 - 1 (\frac{3}{3} \cdot \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 (\frac{\sqrt{3}}{3})})} \cdot \frac{π}{180}$ radians

Step 3.3.8.1.2

Combine.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.2

Apply the distributive property.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.8.3.1

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 1 + 3 (\frac{- \sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.3.2

Cancel the common factor.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 1 + 3 (\frac{- \sqrt{3}}{3})}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.3.3

Rewrite the expression.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 1 - \sqrt{3}}{3 \cdot 1 + 3 (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.4

Multiply $3$ by $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 - \sqrt{3}}{3 \cdot 1 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.5

Simplify the denominator.

Tap for more steps...

Step 3.3.8.5.1

Multiply $3$ by $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 - \sqrt{3}}{3 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.8.5.2.1

Factor $3$ out of $3 \cdot 1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 - \sqrt{3}}{3 + 3 (1) (\frac{\sqrt{3}}{3})}} \cdot \frac{π}{180}$ radians

Step 3.3.8.5.2.2

Cancel the common factor.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 - \sqrt{3}}{3 + 3 \cdot (1 (\frac{\sqrt{3}}{3}))}} \cdot \frac{π}{180}$ radians

Step 3.3.8.5.2.3

Rewrite the expression.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 - \sqrt{3}}{3 + \sqrt{3}}} \cdot \frac{π}{180}$ radians

Step 3.3.8.6

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{3 - \sqrt{3}}{1 - 1 (\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}})} \cdot \frac{π}{180}$ radians

Step 3.3.8.7

Multiply $\frac{3 - \sqrt{3}}{3 + \sqrt{3}}$ by $\frac{3 - \sqrt{3}}{3 - \sqrt{3}}$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}} \cdot \frac{π}{180}$ radians

Step 3.3.8.8

Expand the denominator using the FOIL method.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}} \cdot \frac{π}{180}$ radians

Step 3.3.8.9

Simplify.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.10

Simplify the numerator.

Tap for more steps...

Step 3.3.8.10.1

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.10.2

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.10.3

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.10.4

Add $1$ and $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{{(3 - \sqrt{3})}^{2}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.11

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.12

Expand $(3 - \sqrt{3}) (3 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 3.3.8.12.1

Apply the distributive property.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.12.2

Apply the distributive property.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.12.3

Apply the distributive property.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13

Simplify and combine like terms.

Tap for more steps...

Step 3.3.8.13.1

Simplify each term.

Tap for more steps...

Step 3.3.8.13.1.1

Multiply $3$ by $3$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.2

Multiply $- 1$ by $3$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.3

Multiply $3$ by $- 1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 3.3.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4.2

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4.3

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4.4

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4.5

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.4.6

Add $1$ and $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5

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

Tap for more steps...

Step 3.3.8.13.1.5.1

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5.2

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5.3

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

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.8.13.1.5.4.1

Cancel the common factor.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5.4.2

Rewrite the expression.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.1.5.5

Evaluate the exponent.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.2

Add $9$ and $3$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.13.3

Subtract $3 \sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{12 - 6 \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14

Cancel the common factor of $12 - 6 \sqrt{3}$ and $6$ .

Tap for more steps...

Step 3.3.8.14.1

Factor $6$ out of $12$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{6 \cdot 2 - 6 \sqrt{3}}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.2

Factor $6$ out of $- 6 \sqrt{3}$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.3

Factor $6$ out of $6 (2) + 6 (- \sqrt{3})$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{6 (2 - \sqrt{3})}{6}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.4

Cancel the common factors.

Tap for more steps...

Step 3.3.8.14.4.1

Factor $6$ out of $6$ .

$\frac{3 - \sqrt{3}}{1 - 1 \frac{6 (2 - \sqrt{3})}{6 (1)}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.4.2

Cancel the common factor.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{6 (2 - \sqrt{3})}{6 \cdot 1}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.4.3

Rewrite the expression.

$\frac{3 - \sqrt{3}}{1 - 1 \frac{2 - \sqrt{3}}{1}} \cdot \frac{π}{180}$ radians

Step 3.3.8.14.4.4

Divide $2 - \sqrt{3}$ by $1$ .

$\frac{3 - \sqrt{3}}{1 - 1 (2 - \sqrt{3})} \cdot \frac{π}{180}$ radians

Step 3.4

Apply the distributive property.

$\frac{3 - \sqrt{3}}{1 - 1 \cdot 2 - 1 (- \sqrt{3})} \cdot \frac{π}{180}$ radians

Step 3.5

Multiply $- 1$ by $2$ .

$\frac{3 - \sqrt{3}}{1 - 2 - 1 (- \sqrt{3})} \cdot \frac{π}{180}$ radians

Step 3.6

Multiply $- 1 (- \sqrt{3})$ .

Tap for more steps...

Step 3.6.1

Multiply $- 1$ by $- 1$ .

$\frac{3 - \sqrt{3}}{1 - 2 + 1 \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 3.6.2

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

$\frac{3 - \sqrt{3}}{1 - 2 + \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 3.7

Subtract $2$ from $1$ .

$\frac{3 - \sqrt{3}}{- 1 + \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 4

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

$\frac{3 - \sqrt{3}}{- 1 + \sqrt{3}} \cdot \frac{- 1 - \sqrt{3}}{- 1 - \sqrt{3}} \cdot \frac{π}{180}$ radians

Step 5

Combine fractions.

Tap for more steps...

Step 5.1

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

$\frac{(3 - \sqrt{3}) (- 1 - \sqrt{3})}{(- 1 + \sqrt{3}) (- 1 - \sqrt{3})} \cdot \frac{π}{180}$ radians

Step 5.2

Expand the denominator using the FOIL method.

$\frac{(3 - \sqrt{3}) (- 1 - \sqrt{3})}{1 + \sqrt{3} + \sqrt{3} \cdot - 1 - {\sqrt{3}}^{2}} \cdot \frac{π}{180}$ radians

Step 5.3

Simplify.

$\frac{(3 - \sqrt{3}) (- 1 - \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Expand $(3 - \sqrt{3}) (- 1 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$\frac{3 (- 1 - \sqrt{3}) - \sqrt{3} (- 1 - \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.1.2

Apply the distributive property.

$\frac{3 \cdot - 1 + 3 (- \sqrt{3}) - \sqrt{3} (- 1 - \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.1.3

Apply the distributive property.

$\frac{3 \cdot - 1 + 3 (- \sqrt{3}) - \sqrt{3} \cdot - 1 - \sqrt{3} (- \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2

Simplify and combine like terms.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $3$ by $- 1$ .

$\frac{- 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot - 1 - \sqrt{3} (- \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- 3 - 3 \sqrt{3} - \sqrt{3} \cdot - 1 - \sqrt{3} (- \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.3

Multiply $- \sqrt{3} \cdot - 1$ .

Tap for more steps...

Step 6.2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 3 - 3 \sqrt{3} + 1 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.3.2

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

$\frac{- 3 - 3 \sqrt{3} + \sqrt{3} - \sqrt{3} (- \sqrt{3})}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 6.2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 3 - 3 \sqrt{3} + \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.4.2

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

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

Step 6.2.1.4.3

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

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

Step 6.2.1.4.4

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

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

Step 6.2.1.4.5

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

$\frac{- 3 - 3 \sqrt{3} + \sqrt{3} + {\sqrt{3}}^{1 + 1}}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.4.6

Add $1$ and $1$ .

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

Step 6.2.1.5

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

Tap for more steps...

Step 6.2.1.5.1

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

$\frac{- 3 - 3 \sqrt{3} + \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.5.2

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

$\frac{- 3 - 3 \sqrt{3} + \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.1.5.3

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

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

Step 6.2.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.1.5.4.1

Cancel the common factor.

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

Step 6.2.1.5.4.2

Rewrite the expression.

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

Step 6.2.1.5.5

Evaluate the exponent.

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

Step 6.2.2

Add $- 3$ and $3$ .

$\frac{0 - 3 \sqrt{3} + \sqrt{3}}{- 2} \cdot \frac{π}{180}$ radians

Step 6.2.3

Subtract $3 \sqrt{3}$ from $0$ .

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

Step 6.2.4

Add $- 3 \sqrt{3}$ and $\sqrt{3}$ .

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

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1.1

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 7.1.2

Factor $2$ out of $180$ .

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

Step 7.1.3

Cancel the common factor.

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

Step 7.1.4

Rewrite the expression.

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

Step 7.2

Multiply $\frac{- \sqrt{3}}{- 2}$ by $\frac{π}{90}$ .

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

Step 7.3

Multiply $- 2$ by $90$ .

$\frac{- \sqrt{3} π}{- 180}$ radians

Step 7.4

Dividing two negative values results in a positive value.

$\frac{\sqrt{3} π}{180}$ radians