Find all solutions of the equation Cos(2x) −3 sin(x) + 1 = 0?

Answer 1

Let cos(2x) = 1 - 2sin2(x) Then the expression becomes

1 - 2sin2(x) -3sin(x) + 2 = 0 or sin2(x) + (3/2)sin(x) -1 = 0 Treat sin(x) as a variable in a quadratic

equation & you get

sin(x) = 1/2 or -2 but the second answer if not feasible. So, sin(x) = 1/2

or x = pi/6

Answer 2

use the identity cos(2x)=1-2sin^2(x) which gives us

1-2sin^2(x) -3sin(x)+1=0

-2sin^2(x)-3sin(x) +2=0. Multiply both sides by-1

2sin^2x +3sin(x) -2=0. Factor thus quadratic

(2sinx -1)(sinx+2) =0 this gives us 2 solutions

sinx = 1/2. and sinx =-2.( this is extraneous as sin ranges from 1to-1)

solve sinx= 1/2

x=30 degrees (pi/6) and 150degrees (5pi/6)

Going back in the negative direction we get -210degrees (-7pi/6) and -330degrees(-11pi/6)

Answer 3

There are lots of answers! However, between -2pi and +2pi, the values are (in radians)

x = .5236

x = 2.618

x = -3.6652

x = -5.7596