What is the general solution for 3cos(x/2)-cos(x)=2 ?

The general solution for 3cos(x/2)-cos(x)=2 is x=(2pi)/3+4pin,x=(10pi)/3+4pin,x=4pin

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3cos(x/2)-cos(x)=2

3 cos (\frac{x}{2}) - cos (x) = 2

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn, x = 4 πn

Solution steps

3 cos (\frac{x}{2}) - cos (x) = 2

Subtract

2

from both sides

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

Let:

u = \frac{x}{2}

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

Rewrite using trig identities

- 1 - 2 cos^{2} (u) + 3 cos (u) = 0

Solve by substitution

cos (u) = \frac{1}{2}, cos (u) = 1

cos (u) = \frac{1}{2} : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

cos (u) = 1 : u = 2 πn

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

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

\frac{x}{2} = \frac{5 π}{3} + 2 πn : x = \frac{10 π}{3} + 4 πn

\frac{x}{2} = 2 πn : x = 4 πn

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn, x = 4 πn

tan (x) - csc (2 x) = 0

x, z = sin (x^{2} + y^{2})

- 2 = 2 cos (4 t)

tan (a) = \frac{4}{5}

tan (θ) = \frac{78.48}{196.2}

What is the general solution for 3cos(x/2)-cos(x)=2 ?
The general solution for 3cos(x/2)-cos(x)=2 is x=(2pi)/3+4pin,x=(10pi)/3+4pin,x=4pin