What is the general solution for cos(2x)=1+sin(x) ?

The general solution for cos(2x)=1+sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=2pin,x=pi+2pin

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cos(2x)=1+sin(x)

cos (2 x) = 1 + sin (x)

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

Solution steps

cos (2 x) = 1 + sin (x)

Subtract

1 + sin (x)

from both sides

cos (2 x) - 1 - sin (x) = 0

Rewrite using trig identities

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

Solve by substitution

sin (x) = - \frac{1}{2}, sin (x) = 0

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

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

- 2 = - 5 + 3 tan (θ)

sin (a) = - 1

sin (θ) = \frac{8}{9}

sec (2 x) + tan (2 x) = \frac{1}{2}

\frac{2}{tan ( a ) + cot ( a )} = 2 sin (a)

What is the general solution for cos(2x)=1+sin(x) ?
The general solution for cos(2x)=1+sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=2pin,x=pi+2pin