Problem 69 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\csc \frac{7 \pi}{6}$$

Step-by-Step Solution

Answer

The exact value of the expression \(\csc \frac{7 \pi}{6}\) without using a calculator is -2.

Step 1: Identify the reference angle

The first step is to identify the reference angle for the given angle \(\frac{7 \pi}{6}\). Since this angle is an obtuse, lying in the third quadrant, its reference angle \(\theta\) in the first quadrant can be found by subtracting \(\pi\) (or \(180^\circ\)) from \(\frac{7 \pi}{6}\) i.e \(\frac{7 \pi}{6} - \pi = \frac{\pi}{6}\).

Step 2: Find the sine of reference angle

Now find the sine of the reference angle. Since the referenced angle in step 1 falls in the first quadrant, it's sine value remains positive. We know that \(\sin \frac{\pi}{6} = \frac{1}{2}\) from the known values of sine for common angles.

Step 3: Determine the cosecant of the original angle

Next, recall that cosecant is the reciprocal of the sine function. Therefore, the cosecant of our original angle is the reciprocal of the sine function of the reference angle, or \(\csc \frac{7 \pi}{6} = \frac{1}{\sin \frac{7 \pi}{6}}\). Since \(\sin \frac{7 \pi}{6}\) is equal to -\(\sin \frac{\pi}{6} = -\frac{1}{2}\) (As sine is negative in the third quadrant), the reciprocal of -\(\frac{1}{2}\) is -2. Thus, \(\csc \frac{7 \pi}{6} = -2\).