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

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

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2-3sin(x)-cos(2x)=0

2 - 3 sin (x) - cos (2 x) = 0

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

Solution steps

2 - 3 sin (x) - cos (2 x) = 0

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

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

Combine all the solutions

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

cos (2 θ) = \frac{1}{4}

sec (θ) = \frac{25}{7}

(3 + sin (x))^{2} = 13 - cos^{2} (x)

sec (x) = 3, \frac{3 π}{2} \leq x \leq 2 π

tan (2 x) = cot (3 x)

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