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

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

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

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

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) = sin (x) + 1

Subtract

sin (x) + 1

from both sides

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

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

tan^{2} (x) = 0

tan (θ) = - \frac{1}{3}

tan^{2} (x) + tan (x) - 30 = 0

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

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