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Chapter 5: Problem 15

Solve the equation. $$3 \sec ^{2} x-4=0$$

Short Answer

Expert verified
The solution to the equation is \(x = 2n\pi \pm \arccos(\frac{3}{4})\) where n is any integer.

Step by step solution

01

Isolate the secant term

Start by isolating the \(\sec^2(x)\) term on one side of the equation: Add 4 to both sides of the equation \(3\sec^2(x) - 4 = 0\) to get \(3\sec^2(x) = 4\)
02

Solve for the secant term

Then divide both sides by 3 to get \(\sec^2(x) = \frac{4}{3}\).
03

Apply the secant inverse function

In order to find \(x\), you should reverse the Secant function by applying its inverse. Remember that \(\sec(x) = \frac{1}{\cos(x)}\). So \(\cos(x) = \frac{1}{\sec(x)}\) which becomes \(\cos(x) = \frac{3}{4}\)
04

Solve for x

Applying the inverse \(\cos\) function allows you to find that \(x = \arccos (\frac{3}{4})\). This will provide an initial solution in the range of \(0\) to \(\pi\) (or 0 to 180 degrees), but as the function repeats every \(2\pi\) (or 360 degrees), the full set of solutions is \(x = 2n\pi \pm \arccos(\frac{3}{4})\) where n is an integer.

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Most popular questions from this chapter

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{4} x \cos ^{2} x$$

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing window. Use the graphs to determine whether \(y_{1}=y_{2}\) Explain your reasoning. $$y_{1}=\cos (x+2), \quad y_{2}=\cos x+\cos 2$$

Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$

Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing window. Use the graphs to determine whether \(y_{1}=y_{2}\) Explain your reasoning. $$y_{1}=\sin (x+4), \quad y_{2}=\sin x+\sin 4$$
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