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

The general solution for sin(x)=cos(2x) is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/2+2pin

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sin(x)=cos(2x)

sin (x) = cos (2 x)

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solution steps

sin (x) = cos (2 x)

Subtract

cos (2 x)

from both sides

sin (x) - cos (2 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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos^{3} (x) = cos (x)

2 cos (x) = 1

sin (2 x) sin (x) = cos (x)

2 sec^{2} (x) + tan^{2} (x) - 3 = 0

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

What is the general solution for sin(x)=cos(2x) ?
The general solution for sin(x)=cos(2x) is x= pi/6+2pin,x=(5pi)/6+2pin,x=(3pi)/2+2pin