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

The general solution for cos(2x)-3sin(x)-2=0 is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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

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

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

cos (2 x) - 3 sin (x) - 2 = 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{3 π}{2} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

tan (θ) = - 2 sin (θ)

5 cos (x) = 0

2 sin (x) = cos (2 x)

cos (x) = - \frac{4}{5}

2 cos^{2} (x) + 11 sin (x) = 7

What is the general solution for cos(2x)-3sin(x)-2=0 ?
The general solution for cos(2x)-3sin(x)-2=0 is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin