What is the general solution for 2cos^2(x/3)+3sin(x/3)-3=0 ?

The general solution for 2cos^2(x/3)+3sin(x/3)-3=0 is x= pi/2+6pin,x=(5pi)/2+6pin,x=(3pi)/2+6pin

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2cos^2(x/3)+3sin(x/3)-3=0

2 cos^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3}) - 3 = 0

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{3 π}{2} + 6 πn

Solution steps

2 cos^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3}) - 3 = 0

Rewrite using trig identities

- 1 - 2 sin^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3}) = 0

Solve by substitution

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

sin (\frac{x}{3}) = \frac{1}{2} : x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = 1 : x = \frac{3 π}{2} + 6 πn

Combine all the solutions

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{3 π}{2} + 6 πn

y, cos (x) = x (1 + cot (y))

sec^{2} (x) - 3 = 0

- 4 sin (2 θ) = 0

tan (2 θ) + tan (θ) = 0

2 sin (x) cos (x) = - sin (x)

What is the general solution for 2cos^2(x/3)+3sin(x/3)-3=0 ?
The general solution for 2cos^2(x/3)+3sin(x/3)-3=0 is x= pi/2+6pin,x=(5pi)/2+6pin,x=(3pi)/2+6pin