Problem 40 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\csc \frac{9 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \( \csc \frac{9 \pi}{4} \) is \( \sqrt{2} \)

Step 1: Rewrite the problem in terms of sine function

Since the cosecant function is the reciprocal of the sine function, write it in terms of sine. So, \(csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc \frac{9 \pi}{4} = \frac{1}{\sin \frac{9 \pi}{4}}\).

Step 2: Reduce the angle

The angle \(\frac{9 \pi}{4}\) is more than \(2 \pi\) (or 360 degrees), so it has made more than a full cycle around the unit circle. To simplify, subtract multiples of \(2 \pi\) until the remaining angle is in the range from 0 to \(2 \pi\). In this case, \(\frac{9 \pi}{4} - 2 \pi = \frac{9 \pi}{4} - \frac{8 \pi}{4} = \frac{\pi}{4}\).

Step 3: Use the unit circle to find \(\sin \frac{\pi}{4}\)

On the unit circle, the y-coordinate of the point associated with an angle of \(\frac{\pi}{4}\) or 45 degrees is \(\sin \frac{\pi}{4}\). The y-coordinate is \(\frac{\sqrt{2}}{2}\). Therefore, \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).

Step 4: Substitute the value of \(\sin \frac{\pi}{4}\) into the equation

Substitute the value from step 3 into the equation from step 1: \(\csc \frac{9 \pi}{4} = \frac{1}{\sin \frac{9 \pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}}\).

Step 5: Simplify the fraction

Simplify the fraction by multiplying the numerator and denominator by 2 to remove the complex fraction: \(\csc \frac{9 \pi}{4} = \frac{1*2}{\frac{\sqrt{2}*2}{2}} = \frac{2}{\sqrt{2}}\). By rationalizing the denominator, the exact value of \(\csc \frac{9 \pi}{4}\) is \(\sqrt{2}\).