What is the general solution for 2cos(2x)+4sqrt(3)sin(x)=5 ?

The general solution for 2cos(2x)+4sqrt(3)sin(x)=5 is x= pi/3+2pin,x=(2pi)/3+2pin

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2cos(2x)+4sqrt(3)sin(x)=5

2 cos (2 x) + 43 sin (x) = 5

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

Solution steps

2 cos (2 x) + 43 sin (x) = 5

Subtract

5

from both sides

2 cos (2 x) + 43 sin (x) - 5 = 0

Rewrite using trig identities

- 3 - 4 sin^{2} (x) + 4 sin (x) 3 = 0

Solve by substitution

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Combine all the solutions

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

sin (2 x) + 2 cos^{2} (x) = 0

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

x, y - cos (x) = sin (y)

cos (x) = 0, 0 \leq x < 2 π

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

What is the general solution for 2cos(2x)+4sqrt(3)sin(x)=5 ?
The general solution for 2cos(2x)+4sqrt(3)sin(x)=5 is x= pi/3+2pin,x=(2pi)/3+2pin