What is the general solution for 3sin^2(x)+2=5sin(x) ?

The general solution for 3sin^2(x)+2=5sin(x) is x= pi/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin

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3sin^2(x)+2=5sin(x)

3 sin^{2} (x) + 2 = 5 sin (x)

x = \frac{π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Solution steps

3 sin^{2} (x) + 2 = 5 sin (x)

Solve by substitution

sin (x) = 1, sin (x) = \frac{2}{3}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = \frac{2}{3} : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

sin (θ) = 0.65

2 = - 4 cos (θ)

3 \cdot sin (x) = 3 \cdot cos (x)

12 sin (3 x) = 0

sin (θ) cos (θ) = cos (θ)

What is the general solution for 3sin^2(x)+2=5sin(x) ?
The general solution for 3sin^2(x)+2=5sin(x) is x= pi/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin