Precalculus Examples

$\tan (\frac{7 π}{6} + \frac{3 π}{4})$

Step 1

To write $\frac{7 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\tan (\frac{7 π}{6} \cdot \frac{2}{2} + \frac{3 π}{4})$

Step 2

To write $\frac{3 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\tan (\frac{7 π}{6} \cdot \frac{2}{2} + \frac{3 π}{4} \cdot \frac{3}{3})$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{7 π}{6}$ by $\frac{2}{2}$ .

$\tan (\frac{7 π \cdot 2}{6 \cdot 2} + \frac{3 π}{4} \cdot \frac{3}{3})$

Step 3.2

Multiply $6$ by $2$ .

$\tan (\frac{7 π \cdot 2}{12} + \frac{3 π}{4} \cdot \frac{3}{3})$

Step 3.3

Multiply $\frac{3 π}{4}$ by $\frac{3}{3}$ .

$\tan (\frac{7 π \cdot 2}{12} + \frac{3 π \cdot 3}{4 \cdot 3})$

Step 3.4

Multiply $4$ by $3$ .

$\tan (\frac{7 π \cdot 2}{12} + \frac{3 π \cdot 3}{12})$

Step 4

Combine the numerators over the common denominator.

$\tan (\frac{7 π \cdot 2 + 3 π \cdot 3}{12})$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Multiply $2$ by $7$ .

$\tan (\frac{14 π + 3 π \cdot 3}{12})$

Step 5.2

Multiply $3$ by $3$ .

$\tan (\frac{14 π + 9 π}{12})$

Step 5.3

Add $14 π$ and $9 π$ .

$\tan (\frac{23 π}{12})$

Step 6

The exact value of $\tan (\frac{23 π}{12})$ is $- 2 + \sqrt{3}$ .

Tap for more steps...

Step 6.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$- \tan (\frac{π}{12})$

Step 6.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$- \tan (\frac{π}{4} - \frac{π}{6})$

Step 6.3

Apply the difference of angles identity.

$- \frac{\tan (\frac{π}{4}) - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 6.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- \frac{1 - \tan (\frac{π}{6})}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 6.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$- \frac{1 - \frac{\sqrt{3}}{3}}{1 + \tan (\frac{π}{4}) \tan (\frac{π}{6})}$

Step 6.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$- \frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \tan (\frac{π}{6})}$

Step 6.7

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

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

Step 6.8

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

Tap for more steps...

Step 6.8.1

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

Tap for more steps...

Step 6.8.1.1

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

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

Step 6.8.1.2

Combine.

$- \frac{3 (1 - \frac{\sqrt{3}}{3})}{3 (1 + 1 \frac{\sqrt{3}}{3})}$

Step 6.8.2

Apply the distributive property.

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

Step 6.8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.8.3.1

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

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

Step 6.8.3.2

Cancel the common factor.

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

Step 6.8.3.3

Rewrite the expression.

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

Step 6.8.4

Multiply $3$ by $1$ .

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

Step 6.8.5

Simplify the denominator.

Tap for more steps...

Step 6.8.5.1

Multiply $3$ by $1$ .

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

Step 6.8.5.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.8.5.2.1

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

$- \frac{3 - \sqrt{3}}{3 + 3 (1) \frac{\sqrt{3}}{3}}$

Step 6.8.5.2.2

Cancel the common factor.

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

Step 6.8.5.2.3

Rewrite the expression.

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

Step 6.8.6

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

$- (\frac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \frac{3 - \sqrt{3}}{3 - \sqrt{3}})$

Step 6.8.7

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

$- \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{(3 + \sqrt{3}) (3 - \sqrt{3})}$

Step 6.8.8

Expand the denominator using the FOIL method.

$- \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}$

Step 6.8.9

Simplify.

$- \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 6.8.10

Simplify the numerator.

Tap for more steps...

Step 6.8.10.1

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

$- \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}$

Step 6.8.10.2

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

$- \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}$

Step 6.8.10.3

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

$- \frac{{(3 - \sqrt{3})}^{1 + 1}}{6}$

Step 6.8.10.4

Add $1$ and $1$ .

$- \frac{{(3 - \sqrt{3})}^{2}}{6}$

Step 6.8.11

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

$- \frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{6}$

Step 6.8.12

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

Tap for more steps...

Step 6.8.12.1

Apply the distributive property.

$- \frac{3 (3 - \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 6.8.12.2

Apply the distributive property.

$- \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} (3 - \sqrt{3})}{6}$

Step 6.8.12.3

Apply the distributive property.

$- \frac{3 \cdot 3 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 6.8.13

Simplify and combine like terms.

Tap for more steps...

Step 6.8.13.1

Simplify each term.

Tap for more steps...

Step 6.8.13.1.1

Multiply $3$ by $3$ .

$- \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 6.8.13.1.2

Multiply $- 1$ by $3$ .

$- \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Step 6.8.13.1.3

Multiply $3$ by $- 1$ .

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

Step 6.8.13.1.4

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

Tap for more steps...

Step 6.8.13.1.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

Step 6.8.13.1.4.2

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

Step 6.8.13.1.4.3

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

Step 6.8.13.1.4.4

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

Step 6.8.13.1.4.5

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Step 6.8.13.1.4.6

Add $1$ and $1$ .

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}$

Step 6.8.13.1.5

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

Tap for more steps...

Step 6.8.13.1.5.1

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

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

Step 6.8.13.1.5.2

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

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

Step 6.8.13.1.5.3

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 6.8.13.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.8.13.1.5.4.1

Cancel the common factor.

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Step 6.8.13.1.5.4.2

Rewrite the expression.

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Step 6.8.13.1.5.5

Evaluate the exponent.

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}$

Step 6.8.13.2

Add $9$ and $3$ .

$- \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}$

Step 6.8.13.3

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

$- \frac{12 - 6 \sqrt{3}}{6}$

Step 6.8.14

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

Tap for more steps...

Step 6.8.14.1

Factor $6$ out of $12$ .

$- \frac{6 \cdot 2 - 6 \sqrt{3}}{6}$

Step 6.8.14.2

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

$- \frac{6 \cdot 2 + 6 (- \sqrt{3})}{6}$

Step 6.8.14.3

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

$- \frac{6 (2 - \sqrt{3})}{6}$

Step 6.8.14.4

Cancel the common factors.

Tap for more steps...

Step 6.8.14.4.1

Factor $6$ out of $6$ .

$- \frac{6 (2 - \sqrt{3})}{6 (1)}$

Step 6.8.14.4.2

Cancel the common factor.

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

Step 6.8.14.4.3

Rewrite the expression.

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

Step 6.8.14.4.4

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

$- (2 - \sqrt{3})$

Step 6.8.15

Apply the distributive property.

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

Step 6.8.16

Multiply $- 1$ by $2$ .

$- 2 - - \sqrt{3}$

Step 6.8.17

Multiply $- - \sqrt{3}$ .

Tap for more steps...

Step 6.8.17.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

Step 6.8.17.2

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

$- 2 + \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$