Find the Exact Value (sin((11pi)/6)-cos(pi/6))/(sin(-pi/6)+cos((5pi)/6))

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

Step 1

Simplify the numerator.

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.

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

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

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

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

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

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.

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

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

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.

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

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

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

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.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$1$

Find the Exact Value (sin((11pi)/6)-cos(pi/6))/(sin(-pi/6)+cos((5pi)/6))

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