What is the general solution for cos(2x)=5sin(x)-2 ?

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

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cos(2x)=5sin(x)-2

cos (2 x) = 5 sin (x) - 2

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

Solution steps

cos (2 x) = 5 sin (x) - 2

Subtract

5 sin (x) - 2

from both sides

cos (2 x) - 5 sin (x) + 2 = 0

Rewrite using trig identities

3 - 2 sin^{2} (x) - 5 sin (x) = 0

Solve by substitution

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

sin (x) = - 3 :

No Solution

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan (θ) = 0.48

2 sec (x) tan (x) + 2 sec (x) + tan (x) + 1 = 0

6 tan^{2} (θ) cos (θ) + 3 tan^{2} (θ) = 0

2 sin^{2} (x) - cos (x) = 1, 0 \leq x < 2 π

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

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