Answer:
x = 2 π n_1 + π/2 for n_1 element Z
or x = 2 π n_2 + (5 π)/6 for n_2 element Z or x = 2 π n_3 + π/6 for n_3 element Z
Step-by-step explanation:
Solve for x:
2 cos^2(x) + 3 sin(x) = 3
Write 2 cos^2(x) + 3 sin(x) = 3 in terms of sin(x) using the identity cos^2(x) = 1 - sin^2(x):
-1 + 3 sin(x) - 2 sin^2(x) = 0
The left hand side factors into a product with three terms:
-(sin(x) - 1) (2 sin(x) - 1) = 0
Multiply both sides by -1:
(sin(x) - 1) (2 sin(x) - 1) = 0
Split into two equations:
sin(x) - 1 = 0 or 2 sin(x) - 1 = 0
Add 1 to both sides:
sin(x) = 1 or 2 sin(x) - 1 = 0
Take the inverse sine of both sides:
x = 2 π n_1 + π/2 for n_1 element Z
or 2 sin(x) - 1 = 0
Add 1 to both sides:
x = 2 π n_1 + π/2 for n_1 element Z
or 2 sin(x) = 1
Divide both sides by 2:
x = 2 π n_1 + π/2 for n_1 element Z
or sin(x) = 1/2
Take the inverse sine of both sides:
Answer: x = 2 π n_1 + π/2 for n_1 element Z
or x = 2 π n_2 + (5 π)/6 for n_2 element Z or x = 2 π n_3 + π/6 for n_3 element Z
