Find the Exact Value csc(-(5pi)/6)

Problem

$\csc (-\frac{5\pi }{6})$

Solution

1. Apply the odd function property of the cosecant function, which states that $\csc (-x)=-\csc x$

$\csc (-\frac{5\pi }{6})=-\csc \frac{5\pi }{6}$

2. Identify the reference angle for $\frac{5\pi }{6}$ Since the angle is in the second quadrant, the reference angle is $\pi -\frac{5\pi }{6}=\frac{\pi }{6}$

$\text{Reference Angle}=\frac{\pi }{6}$

3. Determine the sign of the cosecant function in the second quadrant. Since $\sin x$ is positive in the second quadrant, $\csc x$ is also positive.

$\csc \frac{5\pi }{6}=\csc \frac{\pi }{6}$

4. Evaluate the cosecant of the reference angle using the reciprocal identity $\csc x=\frac{1}{\sin x}$

$\csc \frac{\pi }{6}=\frac{1}{\sin \frac{\pi }{6}}$

5. Substitute the known value $\sin \frac{\pi }{6}=\frac{1}{2}$ into the expression.

$\csc \frac{\pi }{6}=\frac{1}{1/2}=2$

6. Combine the results by applying the negative sign from the first step.

$-\csc \frac{5\pi }{6}=-2$

Final Answer

$\csc (-\frac{5\pi }{6})=-2$

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