What is the general solution for sin(x/2)-cos(x)=0 ?

The general solution for sin(x/2)-cos(x)=0 is x= pi/3+4pin,x=(5pi)/3+4pin,x=3pi+4pin

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

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

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = 3 π + 4 πn

Solution steps

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

Let:

u = \frac{x}{2}

sin (u) - cos (2 u) = 0

Rewrite using trig identities

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

Solve by substitution

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

sin (u) = \frac{1}{2} : u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

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

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = 3 π + 4 πn

3 sin (x) = 3 - 3 cos (x)

0 = 1 - 2 cos (x)

- 5 sin^{2} (θ) + 17 cos (θ) + 9 = 4 cos (θ) - 2

(cos (3 θ)) (cos (θ)) + 1 = (sin (3 θ)) (sin (θ))

4 sin (x) - 3 = 0

What is the general solution for sin(x/2)-cos(x)=0 ?
The general solution for sin(x/2)-cos(x)=0 is x= pi/3+4pin,x=(5pi)/3+4pin,x=3pi+4pin