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Chapter 5: Problem 15

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

Short Answer

Expert verified
The solutions to the equation are \(x = \frac{\pi }{6}, \frac{11\pi }{6}, \frac{5\pi }{6}, \frac{7\pi }{6}\)

Step by step solution

01

Isolate sec^2 x Term

To isolate \(\sec ^{2} x\), we can add 4 to both sides of the equation and then divide by 3: \( \sec ^{2} x = \frac{4}{3}\)
02

Convert into Terms of Cosine

Because \(\sec x\) is the reciprocal of \(\cos x\), we can rewrite this equation as \( \cos ^{2} x = \frac{3}{4}\)
03

Take the Square Root

Taking the square root of both sides gives \( \cos x = \pm\sqrt{\frac{3}{4}}\) or \( \cos x = \pm \frac{\sqrt{3}}{2} \)
04

Find Possible Values of x

Referencing the unit circle or a trigonometric table, we can find that \(\cos x = \frac{\sqrt{3}}{2}\) when \(x = \frac{\pi }{6}\) or \(x = \frac{11\pi }{6}\), and \(\cos x = -\frac{\sqrt{3}}{2}\) when \(x = \frac{5\pi }{6}\) or \(x = \frac{7\pi }{6}\).

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