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

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

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4sin^2(x)=4cos(x)+5

4 sin^{2} (x) = 4 cos (x) + 5

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Solution steps

4 sin^{2} (x) = 4 cos (x) + 5

Subtract

4 cos (x) + 5

from both sides

4 sin^{2} (x) - 4 cos (x) - 5 = 0

Rewrite using trig identities

- 1 - 4 cos (x) - 4 cos^{2} (x) = 0

Solve by substitution

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sin (3 x - \frac{π}{5}) = 0 - 1

2 + sin (3 θ) = 3 - sin (3 θ)

sin (2 x) = - cos (3 x) + sin (4 x)

sin (x + \frac{π}{2}) = - cos (x)

sin (x) = (\frac{3}{5})

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