What is the general solution for 4sin(x/2)+2cos(x)=3 ?

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

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4sin(x/2)+2cos(x)=3

4 sin (\frac{x}{2}) + 2 cos (x) = 3

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

Solution steps

4 sin (\frac{x}{2}) + 2 cos (x) = 3

Subtract

3

from both sides

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

Let:

u = \frac{x}{2}

4 sin (u) + 2 cos (2 u) - 3 = 0

Rewrite using trig identities

- 1 + 4 sin (u) - 4 sin^{2} (u) = 0

Solve by substitution

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

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

Combine all the solutions

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

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

cos (x) = \frac{1}{5}

sin (t) = \frac{3}{2}

sin (θ) = \frac{15}{17}

sin (θ) = cos (2 θ)

13 sin (x) + 6 = sin (x)

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