Find the general solution for x if cos2x + sin3x = sinx

1 answer

Nice question !

cos2x + sin3x = sinx
let's change everything to sinx

(1 - 2sin^2 x) + (3sinx - 4sin^3 x) - sinx = 0
4sin^3 x + 2sin^2 x - 2sinx -1 = 0
2sin^2 x (2sinx + 1) - 1(2sinx + 1) = 0
(2sinx + 1) (2sin^2x -1) = 0

sinx = -1/2 or sin^2 x = 1/2

for sinx = -1/2
x = 7π/6 or 11π/6 --------- ( 210° or 330°)

for sin^2 x = 1/2
sinx = ± 1/√2
x = π/4 , 3π/4 , 5π/4 , 7π/4 ------- ( 45°, 135°, 225° , 315°)

for general solutions take each of the above answers and add 2kπ

e.g. 7π/6 + 2kπ , where k is an integer.
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