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Chapter 5: Problem 18

Solve the equation. $$\sin ^{2} x=3 \cos ^{2} x$$

Short Answer

Expert verified
The solutions to the equation \(\sin^2{x} = 3\cos^2{x}\) are \(x = 0, \pi\).

Step by step solution

01

Rewrite the equation

Rewrite the given equation in terms of \(\cos^2{x}\), as follows: \(\cos^2{x} = \frac{\sin^2{x}}{3}\).
02

Apply Pythagorean Identity

Since the Pythagorean Identity is \(\sin^2{x} + \cos^2{x} = 1\), this can be rearranged to give \(\cos^2{x} = 1 - \sin^2{x}\) or \(\sin^2{x} = 1 - \cos^2{x}\). Substitute this into the equation from Step 1 to get \(\frac{1 - \cos^2{x}}{3} = \cos^2{x}\), simplifying which gives \(\cos^4{x}-\frac{1}{3}\cos^2{x}-\frac{2}{3} = 0\).
03

Substitute \(\cos^2{x} = t \)

Let \(\cos^2{x} = t\). This transforms the equation from Step 2 into a simpler quadratic equation: \(t^2 - \frac{1}{3}t - \frac{2}{3} = 0\).
04

Solve the quadratic equation

Use the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve the equation from Step 3. This gives two solutions: \(t_1 = 1\) and \(t_2 = - \frac{2}{3}\).
05

Solve for \(x\) and exclude invalid solution

By substituting the solutions from Step 4 back into the equation \(\cos^2{x} = t\), solve for \(x\). Keep in mind that since \(\cos^2{x}\) is a square, it cannot be negative. Therefore, the solution \(t = - \frac{2}{3}\) is invalid. The remaining valid solution is \(\cos^2{x} = 1\), which gives \(x = 0, \pi\).

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Most popular questions from this chapter

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