What is the general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 ?

The general solution for cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2 is x=(3pi)/2 ,x=(7pi)/6

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cos(2x)=2+3sin(x), pi/2 <= x<= (3pi)/2

cos (2 x) = 2 + 3 sin (x), \frac{π}{2} \leq x \leq \frac{3 π}{2}

x = \frac{3 π}{2}, x = \frac{7 π}{6}

Solution steps

cos (2 x) = 2 + 3 sin (x), \frac{π}{2} \leq x \leq \frac{3 π}{2}

Subtract

2 + 3 sin (x)

from both sides

cos (2 x) - 2 - 3 sin (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, \frac{π}{2} \leq x \leq \frac{3 π}{2} : x = \frac{3 π}{2}

sin (x) = - \frac{1}{2}, \frac{π}{2} \leq x \leq \frac{3 π}{2} : x = \frac{7 π}{6}

Combine all the solutions

x = \frac{3 π}{2}, x = \frac{7 π}{6}

tan (θ) = (\frac{3}{8})

29 = \frac{1}{2} (5.6) (7.1) sin (x)

\frac{10.4}{sin ( x )} = \frac{12}{sin ( 6 2 ^{\circ} )}

csc (θ) = - 2, 0^{\circ} \leq θ \leq 36 0^{\circ}

sin (B) = \frac{12}{13}

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