for all values of theta where 0degrees<theta<360degrees
2sin^2 theta -3sintheta = -1
This is a quadratic which factors ...
2sin^2 Ø - 3sinØ + 1 = 0
(2sinØ - 1)(sinØ - 1) = 0
sinØ = 1/2 or sinØ = 1
Ø = 30° or 150° or Ø = 90°
To solve the equation 2sin^2(theta) - 3sin(theta) = -1, we can follow these steps:
Step 1: Rewrite the equation:
2sin^2(theta) - 3sin(theta) + 1 = 0
Step 2: Factor the quadratic equation:
(2sin(theta) - 1)(sin(theta) - 1) = 0
Step 3: Solve for sin(theta) separately:
2sin(theta) - 1 = 0 --> sin(theta) = 1/2
sin(theta) - 1 = 0 --> sin(theta) = 1
Step 4: Find the values of theta that satisfy the equations using inverse sine function (sin^(-1)):
For sin(theta) = 1/2:
theta = sin^(-1)(1/2)
theta = 30 degrees or 150 degrees (in the given range of 0 degrees < theta < 360 degrees)
For sin(theta) = 1:
theta = sin^(-1)(1)
theta = 90 degrees (in the given range of 0 degrees < theta < 360 degrees)
So, the values of theta that satisfy the equation 2sin^2(theta) - 3sin(theta) = -1, in the given range 0 degrees < theta < 360 degrees, are 30 degrees, 90 degrees, and 150 degrees.