Find all solutions for the following equations. Write your answers in radians using exact values.

Question

cos 2x + 3 cos x = -2

Answer 1

IF the problem is

cos^2(x) + 3cosx =-2

then

let y = cosx

y^2 + 3y -2 = 0

(y+1)(y+2) =0

y = -1, -2

cosx = -1, -2 (ignore -2 as cosines are never less than -1)

x = pi +2npi radians, where n= any integer

x = (1+2n)pi radians

but maybe you did mean

cos2x + 3cosx = -2

then

cos^2(x) - sin^2(x) + 3cosx = -2

cos^2(x) +3cosx +2 = sin^2(x)

cos^2(x) + sin^2(x) + 3cosx +2 = 2sin^2(x)

3cosx +3 = 2sin^2(x)= 2(1-cosx)/2

3cosx +3 = 1-cosx

4cosx = -2

cosx = -1/2

x = 120 or 240 degrees + 360n

= 2pi/3 +2npi = (2/3 + 2n)pi

or 4pi/3 + 2npi = (4/3 + 2n)pi where n = any integer

x = (2/3+2n)pi, (4/3 +2n)pi