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

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

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cos(2x)=-sin(x)

cos (2 x) = - sin (x)

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

Solution steps

cos (2 x) = - sin (x)

Subtract

- sin (x)

from both sides

cos (2 x) + sin (x) = 0

Rewrite using trig identities

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

Solve by substitution

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

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

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

3 cot (t) - 1 = 0, 0 \leq t \leq 2 π

10 cos (c) + 1 = cos (c) - 1

tan (x - \frac{π}{3}) = 1

3 sin (x) + cos (x) = 3

1 = sin (2 t)

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