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Chapter 5: Problem 74

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\csc ^{2} x-5 \csc x=0$$

Short Answer

Expert verified
The solutions of the given equation in the interval \([0,2 \pi)\) are \(x=0, \pi, \sin^{-1}\left(\frac{1}{5}\right), \pi - \sin^{-1}\left(\frac{1}{5}\right)\)

Step by step solution

01

Rearrange the Equation

Write the equation in the standard form of a quadratic equation. Let \(u = \csc(x)\), the equation then becomes \(u^2 - 5u = 0\). This will simplify the process of finding roots.
02

Solve for Inverse Function

Factor out the equation to find the roots. Solve \(u^2 - 5u = 0\), which results in \(u(u - 5) = 0\). Hence, the roots of the equation are \(u = 0\) and \(u = 5\), i.e., \(\csc(x) = 0\) and \(\csc(x) = 5\).
03

Find Corresponding Angles

Convert back to the sine function and find the corresponding angles which satisfy the equations \(\sin(x) = 0\) and \(\sin(x) = \frac{1}{5}\). The solutions are \(x=0, \pi\) for the first equation and \(x=\sin^{-1}\left(\frac{1}{5}\), \( \pi - \sin^{-1}\left(\frac{1}{5}\right)\) for the second equation.
04

Check the Interval

Verify that each solution lies within the interval \([0,2 \pi)\). All the obtained solutions are within this range.

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Most popular questions from this chapter

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$3 \tan ^{2} x+4 \tan x-4=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$$

Find the exact value of each expression. (a) \(\sin \left(135^{\circ}-30^{\circ}\right)\) (b) \(\sin 135^{\circ}-\cos 30^{\circ}\)

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in minutes) can be modeled by \(h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)\) The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times?

Write the expression as the sine, cosine, or tangent of an angle. $$\cos \frac{\pi}{7} \cos \frac{\pi}{5}-\sin \frac{\pi}{7} \sin \frac{\pi}{5}$$
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