What is the general solution for sin(x/2)= 1/2 sin(x) ?

The general solution for sin(x/2)= 1/2 sin(x) is x=4pin,x=2pi+4pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(x/2)= 1/2 sin(x)

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

x = 4 πn, x = 2 π + 4 πn

Solution steps

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

Subtract

\frac{1}{2} sin (x)

from both sides

sin (\frac{x}{2}) - \frac{1}{2} sin (x) = 0

Simplify

sin (\frac{x}{2}) - \frac{1}{2} sin (x) : \frac{2 sin ( \frac{x}{2} ) - sin ( x )}{2}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Let:

u = \frac{x}{2}

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

Rewrite using trig identities

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

Factor

2 sin (u) - 2 cos (u) sin (u) : - 2 sin (u) (cos (u) - 1)

- 2 sin (u) (cos (u) - 1) = 0

Solving each part separately

sin (u) = 0 or cos (u) - 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

cos (u) - 1 = 0 : u = 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

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

x = 4 πn, x = 2 π + 4 πn

tan (θ) = \frac{- 3}{3}

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

sec^{2} (x) + 2 sec (x) - 8 = 0

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

sin (x) = \frac{7}{15}

What is the general solution for sin(x/2)= 1/2 sin(x) ?
The general solution for sin(x/2)= 1/2 sin(x) is x=4pin,x=2pi+4pin