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

The general solution for 4cos^2(x)=5+4sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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

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

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Subtract

5 + 4 sin (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

cos^{2} (x) - sin (x) = 1

sec^{2} (x) - 5 tan (x) = - 3

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

sin (x + \frac{π}{4}) = \frac{3}{2}

What is the general solution for 4cos^2(x)=5+4sin(x) ?
The general solution for 4cos^2(x)=5+4sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin