"What is the solution of cos⁡2x−cos⁡x=0 in the..." was answered by our community

alesterp

Answered question

2021-02-13

What is the solution of

\cos 2 x - \cos x = 0 in the interval [0, 2 π)

tafzijdeq

Skilled2021-02-14Added 92 answers

Let use the trigoniometric 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 o r 2 \cos x + 1 = 0

Solve both equations to consine:

\cos x = 1 o r \cos x = - \frac{1}{2}

\cos x = 1

when

x = 0

x = - \frac{1}{2}

when

x = π - π / 3 = (2 π) / 3 o r x = π + π / 3 = (4 π) / 3

Aswer:

x = 0, (\frac{2 π}{3}), (4 π) / 3

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