Solve for theta over the interval 0 to 360

Question

cos 2theta + 3 sin theta = 0

Answer 1

Let x = theta

 

cos(2x) + 3sinx = 0

 

 

Use the double angle identity.  We want the identity in terms of sine.

 

cos(2x) = cos²(x) - sin²(x)

           = 1 - sin²(x) - sin²(x)

           = 1 - 2sin²(x)

 

 

Substitute this identity into the equation.

 

1 - 2sin²(x) + 3sin(x) = 0

 

-2sin²(x) + 3sin(x) + 1 = 0

 

 

We can use the quadratic formula to solve for sin(x).

 

sin(x) = (-3 ± √(9 - 4(-2))) / -4

sin(x) = (-3 ± √(17)) / -4

sin(x) = (-3 ± 4.12) / -4

 

sin(x) = (-3 + 4.12) / -4                  and                sin(x) =  (-3 - 4.12) / -4

sin(x) = -0.28                                 and                 sin(x) = 1.78

 

x = 16.26                                      

 

Sine is negative in the 3rd and 4th quadrant.

 

x = 180 + 16.26 = 196.26

x = 360 - 16.26 = 343.75

 

 

The solutions are

 

x = 196.26

x = 343.75