Algebra Examples

$\frac{\sin (\frac{11 π}{6}) - \cos (\frac{π}{6})}{\sin (- \frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{- \sin (\frac{π}{6}) - \cos (\frac{π}{6})}{\sin (- \frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{- \frac{1}{2} - \cos (\frac{π}{6})}{\sin (- \frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 1.3

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

$\frac{- \frac{1}{2} - \frac{\sqrt{3}}{2}}{\sin (- \frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{- 1 - \sqrt{3}}{2}}{\sin (- \frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{- 1 - \sqrt{3}}{2}}{\sin (\frac{5 π}{3}) + \cos (\frac{5 π}{6})}$

Step 2.2

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

$\frac{\frac{- 1 - \sqrt{3}}{2}}{- \sin (\frac{π}{3}) + \cos (\frac{5 π}{6})}$

Step 2.3

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

$\frac{\frac{- 1 - \sqrt{3}}{2}}{- \frac{\sqrt{3}}{2} + \cos (\frac{5 π}{6})}$

Step 2.4

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 second quadrant.

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

Step 2.5

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

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

Step 2.6

Subtract $\frac{\sqrt{3}}{2}$ from $- \frac{\sqrt{3}}{2}$ .

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

Step 3

Combine $- 2$ and $\frac{\sqrt{3}}{2}$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Reduce the expression $\frac{- 2 \sqrt{3}}{2}$ by cancelling the common factors.

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Step 4.1.1

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

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

Step 4.1.2

Factor $2$ out of $2$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

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

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $1$ and $- 1$ .

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Step 6.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.2

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Multiply $2$ by $3$ .

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

Step 10

Apply the distributive property.

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Add $1$ and $1$ .

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

Step 13

Simplify each term.

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Step 13.1

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

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Step 13.1.1

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

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

Step 13.1.2

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

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

Step 13.1.3

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

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

Step 13.1.4

Cancel the common factor of $2$ .

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Step 13.1.4.1

Cancel the common factor.

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

Step 13.1.4.2

Rewrite the expression.

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

Step 13.1.5

Evaluate the exponent.

$- \frac{- \sqrt{3} - 1 \cdot 3}{6}$

Step 13.2

Multiply $- 1$ by $3$ .

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

Step 14

Simplify with factoring out.

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Step 14.1

Factor $- 1$ out of $- \sqrt{3}$ .

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

Step 14.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 14.3

Factor $- 1$ out of $- (\sqrt{3}) - 1 (3)$ .

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

Step 14.4

Simplify the expression.

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Step 14.4.1

Rewrite $- (\sqrt{3} + 3)$ as $- 1 (\sqrt{3} + 3)$ .

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

Step 14.4.2

Move the negative in front of the fraction.

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

Step 14.4.3

Multiply $- 1$ by $- 1$ .

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

Step 14.4.4

Multiply $\frac{\sqrt{3} + 3}{6}$ by $1$ .

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

Step 15

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.78867513 \dots$