Question

how can i find all the exact solutions to the equation : 2cos^2x + 3sinx = 3

the solutions have to be between [0,2pi)

Answer 1

assuming that 2cos^2x means 2(cos(x))^2
substitute using
(cos(x))^2 = 1 - (sin(x))^2
2(1 - (sin(x))^2) + 3sin(x) = 3
2 - 2(sin(x))^2 + 3sin(x) = 3
for clarity, let u = sin(x)
2 - 2u^2 + 3u = 3
rearranging...
2u^2 - 3 u + 1 = 0
Solve this quadratic to find the values of sin x that are solutions. Finally, determine the angles in the range that have that value of sin(x).