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Chapter 5: Problem 20

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

Short Answer

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

Step by step solution

01

Write Everything in Terms of Sine

Using the fundamental identity of trigonometry \(sin^2(x) + cos^2(x) = 1\), solve for \(cos^2(x)\) . This gives \(cos^2(x) = 1 - sin^2(x)\) . Substitute this into the equation to get \(sin^2(x) = 3(1-sin^2(x))\)
02

Simplify and Rearrange

Simplify the equation by multiplying and rearranging. This gives the quadratic equation \(4sin^2(x) -1 = 0\)
03

Solve the Quadratic Equation

Now solve the quadratic equation for \(sin(x)\). Doing this gives two results, \(sin(x) = 1/2\) or \(sin(x) = -1/2\)
04

Find the Values of $x$ Where the Equation is True

Solving for all solutions where the equation is true over \(0 \leq x \leq 2\pi\), find that \(x = \pi/6, 5\pi/6, 7\pi/6, 11\pi/6 \) are the solutions.

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Most popular questions from this chapter

Find the exact value of the expression. $$\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}$$

Determine whether the statement is true or false. Justify your answer. If you correctly solve a trigonometric equation to the statement \(\sin x=3.4\), then you can finish solving the equation by using an inverse function.

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