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=(5pi)/6 ,x= pi/2

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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{5 π}{6}, x = \frac{π}{2}

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

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

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

Combine all the solutions

x = \frac{5 π}{6}, x = \frac{π}{2}

3 sin (x) cos (x) = 0

\frac{1}{2} sin (t) - \frac{1}{2} = - 1

20 sin^{2} (x) - 21 sin (x) + 4 = 0

4 tan (x) + 1 = 0

3 tan (θ) - 4 = tan (θ) - 2

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=(5pi)/6 ,x= pi/2