Expert Answer to "How do you find exact solutions of cos⁡2x−cos⁡x=0..."

Jaya Legge

Answered question

2021-11-08

How do you find exact solutions of

\cos 2 x - \cos x = 0

in the interval

[0, 2 π)

Tasneem Almond

Skilled2021-11-09Added 91 answers

Given:

\cos 2 x - \cos x = 0

Let us use the trigonometric identity

\cos 2 x = 2 \cos^{2} x - 1 :

2 \cos^{2} x - 1 - \cos x = 0

Factorize:

(\cos x - 1) (2 \cos x + 1) = 0

Use the zero product property:

\cos x - 1 = 0 or 2 \cos x + 1 = 0

Solve both equations to cosine:

\cos x = 1 or \cos x = - \frac{1}{2}

\cos x = 1

when x=0

\cos x = - \frac{1}{2} w h e n x = π - \frac{π}{3} = \frac{2 π}{3} or x = π + \frac{π}{3} = \frac{4 π}{3}

.
Answer

x = 0, \frac{2 π}{3}, \frac{4 π}{3}

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