Algebra Examples

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

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.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

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 $2 π$ .

$\frac{\frac{- 1 - \sqrt{3}}{2}}{\sin (\frac{11 π}{6}) + \cos (\frac{5 π}{6})}$

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.

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

Step 2.3

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

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

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.

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 3

Cancel the common factor of $\frac{- 1 - \sqrt{3}}{2}$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$1$