What is the general solution for tan(x)=3tan(1/2 x) ?

The general solution for tan(x)=3tan(1/2 x) is x=2pin,x=(5pi)/3+2pin,x= pi/3+2pin

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tan(x)=3tan(1/2 x)

tan (x) = 3 tan (\frac{1}{2} x)

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

Solution steps

tan (x) = 3 tan (\frac{1}{2} x)

Subtract

3 tan (\frac{1}{2} x)

from both sides

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

Let:

u = \frac{x}{2}

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

Rewrite using trig identities

\frac{- tan ( u ) + 3 tan ^{3} ( u )}{1 - tan ^{2} ( u )} = 0

Solve by substitution

tan (u) = 0, tan (u) = - \frac{3}{3}, tan (u) = \frac{3}{3}

tan (u) = 0 : u = πn

tan (u) = - \frac{3}{3} : u = \frac{5 π}{6} + πn

tan (u) = \frac{3}{3} : u = \frac{π}{6} + πn

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = πn : x = 2 πn

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

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

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

cos (t) = \frac{- 5}{13}, \frac{1}{2} π < t < π

31.91 cos (x) = 20.48

7 sin (x) + 7 sin (x) tan (x) = 0

0 = cos (20 t) - cos (20.02 t)

2 tan (2 x - 10) = 3.4641

What is the general solution for tan(x)=3tan(1/2 x) ?
The general solution for tan(x)=3tan(1/2 x) is x=2pin,x=(5pi)/3+2pin,x= pi/3+2pin