Solve 2 cos 2 ( t ) + 3 cos ( t ) + 1 = 0 for all solutions 0 ≤ t < 2 π

Answer 1

cos 2t = 2cos²t-1

This identity will convert your equation into a quadratic equation in cos t which will allow you to solve the equation as required.

Answer 2

Cos (2t) = 2cos²(t) -1 Refer to the webpage (https://www.purplemath.com/modules/idents.htm )for all important trigonometric identities,

2 cos 2 ( t ) + 3 cos ( t ) + 1 = 0

Replace cos 2 ( t ) with 2cos²(t) -1.

You will get 4cos²(t) +3 cos(t) -1 = 0

simplify to get (4cos(t) - 1)*(cos(t) + 1) = 0

Therefore, cos(t) = -1 or cos(t) = 1/4.

Therefore potential solutions in 0 ≤ t < 2 π would be

t = π

t = cos^-1(1/4)

t = 2π - cos^-1(1/4)