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Chapter 5: Problem 19

Solve the equation. $$4 \cos ^{2} x-1=0$$

Short Answer

Expert verified
The solutions for the equation are \(x = \pi/3, 5\pi/3, 2\pi/3, 4\pi/3\)

Step by step solution

01

Rewrite the equation

The first step is to express the equation in a more convenient form. This can be done by isolating \(\cos ^{2} x\), by adding 1 to both sides and then dividing by 4. This results in: \(\cos ^{2} x = 0.25\) or \(\cos ^{2} x = 1/4\)
02

Extract the square root

Next, take the square root of both sides of the equation to solve for \(|\cos x|\). Keep in mind that the square root of a position number can be positive or negative. Therefore, we have \(|\cos x| = \pm \sqrt{1/4} = \pm 0.5\)
03

Solve for the x-values

Now that we have the numerical value of \(|\cos x|\), we can solve for \(x\). Using the unit circle or a trigonometric table, we can find \(x\) such that it satisfies \(\cos x = 0.5\) or \(\cos x = -0.5\). For an angle in the range of [0,2蟺], the solutions to the equation are \(x = \pi/3, 5\pi/3\) for \(\cos x = 0.5\), and \(x = 2\pi/3, 4\pi/3\) for \(\cos x = -0.5\).

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Most popular questions from this chapter

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Find the exact values of the sine, cosine, and tangent of the angle. $$15^{\circ}$$

Fill in the blank. \(\sin (u+v)=\)_____

Fill in the blank. \(\cos (u+v)=\)_____

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