Scroll to Top

Virtual Math Learning Center

Virtual Math Learning Center Texas A&M University Virtual Math Learning Center
tags:
Cosine Factoring Sine Trigonometry Trigonometry Series

Solving Trigonometric Equations Exercise 2

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem: Solve \(2\sin{x}\cos{x} = \sqrt{2}\cos{x}\) for \(0\leq x \leq 2\pi.\)

Solution: \(x=\dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4},\dfrac{3\pi}{2}\)

Solution Method: A tempting first step here is to divide by a \(\cos{x}\) on both sides. But if we do that, we’ll be losing a solution to the equation! (Think about \(x^2=3x\). We can clearly see that \(x=0\) is a solution to the equation. But if we divide by x, we may think \(x=3\) is the only solution.)

So instead of dividing by \(\cos{x}\) as out first step, we’re going to move everything over to one side and set it equal to zero. \[\begin{aligned} 2\sin{x}\cos{x} &= \sqrt{2}\cos{x} \\ 2\sin{x}\cos{x} - \sqrt{2}\cos{x} &= 0 \end{aligned}\] Then I can factor out a \(\cos{x}\), \[\begin{aligned} 2\sin{x}\cos{x} - \sqrt{2}\cos{x} &= 0 \\ \cos{x}(2\sin{x} - \sqrt{2}) &= 0 \end{aligned}\] So I have two factors I can set equal to 0 and solve for x.

2 \[\begin{aligned} \cos{x} = 0 \end{aligned}\] Zero is a quadrantal value and thinking back to the unit circle, I know \(x=\frac{\pi}{2}, \frac{3\pi}{2}\) satisfy \(\cos{x}=0\). \[\begin{aligned} 2\sin{x}-\sqrt{2} &= 0 \\ 2\sin{x} = \sqrt{2} \\ \sin{x} = \frac{\sqrt{2}}{2} \end{aligned}\] The reference angle that produces sine equal to \(\frac{\sqrt{2}}{2}\) is \(\frac{\pi}{4}\). The sine value is also positive, so I’m in Q1 or Q2 (by ASTC). So the x values satisfying the equation are \(x=\frac{\pi}{4}, \frac{3\pi}{4}\).

So we have a total of four x-values satisfying \(2\sin{x}\cos{x} = \sqrt{2}\cos{x}\), \(x=\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}.\)

See more videos from this section

Prev Video Next Video Return to Section