How do you solve $2 + 3cos2x = cosx$ ?

Question

2919 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-13T18:09:31

Replace cos 2x by $(2cos^2 x - 1)$
f(x) = 2 + 3(2cos^2 x - 1) - cos x = 0. Call cos x = t.

6t^2 - t - 1 = 0. Solve by the formula
$D = d^2 = 1 + 24 = 25 -> d = +- 5$

$t = 1/12 + 5/12 = 6/12 = 1/2$

$t = 1/12 - 5/12 = -4/12 = -1/3$
Use the trig unit circle as proof:

$cos x = t = 1/2 -> x = pi/3 and x = 5pi/3$

$cos x = -1/3 -> x = 109.47 and x = 250.53 deg$

How do you solve 2 + 3cos2x = cosx2 + 3cos2x = cosx?