Precalculus Examples

$\tan (\frac{11 π}{6} + \frac{3 π}{4})$

Step 1

To write $\frac{11 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\tan (\frac{11 π}{6} \cdot \frac{2}{2} + \frac{3 π}{4})$

Step 2

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

$\tan (\frac{11 π}{6} \cdot \frac{2}{2} + \frac{3 π}{4} \cdot \frac{3}{3})$

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 $\frac{11 π}{6}$ by $\frac{2}{2}$ .

$\tan (\frac{11 π \cdot 2}{6 \cdot 2} + \frac{3 π}{4} \cdot \frac{3}{3})$

Step 3.2

Multiply $6$ by $2$ .

$\tan (\frac{11 π \cdot 2}{12} + \frac{3 π}{4} \cdot \frac{3}{3})$

Step 3.3

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

$\tan (\frac{11 π \cdot 2}{12} + \frac{3 π \cdot 3}{4 \cdot 3})$

Step 3.4

Multiply $4$ by $3$ .

$\tan (\frac{11 π \cdot 2}{12} + \frac{3 π \cdot 3}{12})$

Step 4

Combine the numerators over the common denominator.

$\tan (\frac{11 π \cdot 2 + 3 π \cdot 3}{12})$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Multiply $2$ by $11$ .

$\tan (\frac{22 π + 3 π \cdot 3}{12})$

Step 5.2

Multiply $3$ by $3$ .

$\tan (\frac{22 π + 9 π}{12})$

Step 5.3

Add $22 π$ and $9 π$ .

$\tan (\frac{31 π}{12})$

Step 6

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\tan (\frac{7 π}{12})$

Step 7

The exact value of $\tan (\frac{7 π}{12})$ is $- \sqrt{7 + 4 \sqrt{3}}$ .

Tap for more steps...

Step 7.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\tan (\frac{\frac{7 π}{6}}{2})$

Step 7.2

Apply the tangent half-angle identity.

$\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$

Step 7.3

Change the $\pm$ to $-$ because tangent is negative in the second quadrant.

$- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$ .

Tap for more steps...

Step 7.4.1

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

$- \sqrt{\frac{1 - - \cos (\frac{π}{6})}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.3

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

Tap for more steps...

Step 7.4.3.1

Multiply $- 1$ by $- 1$ .

$- \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$- \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.4

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

$- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.5

Combine the numerators over the common denominator.

$- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}$

Step 7.4.6

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

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

Step 7.4.7

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 7.4.8

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

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

Step 7.4.9

Combine the numerators over the common denominator.

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

Step 7.4.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.4.11

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.4.11.1

Cancel the common factor.

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

Step 7.4.11.2

Rewrite the expression.

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

Step 7.4.12

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Step 7.4.13

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Step 7.4.14

Expand the denominator using the FOIL method.

$- \sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}}$

Step 7.4.15

Simplify.

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

Step 7.4.16

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

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

Step 7.4.17

Expand $(2 + \sqrt{3}) (2 + \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 7.4.17.1

Apply the distributive property.

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

Step 7.4.17.2

Apply the distributive property.

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

Step 7.4.17.3

Apply the distributive property.

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

Step 7.4.18

Simplify and combine like terms.

Tap for more steps...

Step 7.4.18.1

Simplify each term.

Tap for more steps...

Step 7.4.18.1.1

Multiply $2$ by $2$ .

$- \sqrt{4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}$

Step 7.4.18.1.2

Move $2$ to the left of $\sqrt{3}$ .

$- \sqrt{4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 7.4.18.1.3

Combine using the product rule for radicals.

$- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}}$

Step 7.4.18.1.4

Multiply $3$ by $3$ .

$- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}}$

Step 7.4.18.1.5

Rewrite $9$ as $3^{2}$ .

$- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}}$

Step 7.4.18.1.6

Pull terms out from under the radical, assuming positive real numbers.

$- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3}$

Step 7.4.18.2

Add $4$ and $3$ .

$- \sqrt{7 + 2 \sqrt{3} + 2 \sqrt{3}}$

Step 7.4.18.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$- \sqrt{7 + 4 \sqrt{3}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{7 + 4 \sqrt{3}}$

Decimal Form:

$- 3.73205080 \dots$