/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Find the exact value of each tri... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 40

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

Short Answer

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

Step by step solution

01

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, \(\csc \frac{9 \pi}{4} = \frac{1}{\sin \frac{9 \pi}{4}}\).
02

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, \(\frac{9 \pi}{4} - 2 \pi = \frac{9 \pi}{4} - \frac{8 \pi}{4} = \frac{\pi}{4}\).
03

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, \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
04

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

Substitute the value from step 3 into the equation from step 1: \(\csc \frac{9 \pi}{4} = \frac{1}{\sin \frac{9 \pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}}\).
05

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 \(\sqrt{2}\).

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Most popular questions from this chapter

A clock with an hour hand that is 15 inches long is hanging on a wall. At noon, the distance between the tip of the hour hand and the ceiling is 23 inches. At 3 P.M., the distance is 38 inches; at 6 P.M., 53 inches; at 9 P.M., 38 inches; and at midnight the distance is again 23 inches. If \(y\) represents the distance between the tip of the hour hand and the ceiling \(x\) hours after noon, make a graph that displays the information for \(0 \leq x \leq 24\)

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made an error because the angle I drew in standard position exceeded a straight angle.

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to a decimal in degrees. Round your answer to two decimal places. $$65^{\circ} 45^{\prime} 20^{\prime \prime}$$

Will help you prepare for the material covered in the next section. a. Graph \(y=\cos x\) for \(0 \leq x \leq \pi\) b. Based on your graph in part (a), does \(y=\cos x\) have an inverse function if the domain is restricted to \([0, \pi] ?\) Explain your answer. c. Determine the angle in the interval \([0, \pi]\) whose cosine is \(-\frac{\sqrt{3}}{2} .\) Identify this information as a point on your graph in part (a).
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