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Popular >

sin(x)=tan(x/2)

Solution

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

Solution

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

+1

Solution steps

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

Subtract

tan (\frac{x}{2})

from both sides

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

Let:

u = \frac{x}{2}

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

- sin (u) + 2 cos^{2} (u) sin (u) : sin (u) (2 cos (u) + 1) (2 cos (u) - 1)

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

Solving each part separately

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

2 cos (u) + 1 = 0 : u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

2 cos (u) - 1 = 0 : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 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

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

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

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

\frac{x}{2} = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{2} + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

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