What is the general solution for tan(x/2)=sqrt(3),0<= x<2pi ?

The general solution for tan(x/2)=sqrt(3),0<= x<2pi is x=(2pi)/3

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tan(x/2)=sqrt(3),0<= x<2pi

tan (\frac{x}{2}) = 3, 0 \leq x < 2 π

x = \frac{2 π}{3}

Degrees

x = 12 0^{\circ}

Solution steps

tan (\frac{x}{2}) = 3, 0 \leq x < 2 π

General solutions for

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

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{π}{3} + πn

\frac{x}{2} = \frac{π}{3} + πn

Solve

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

\frac{x}{2} = \frac{π}{3} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{3} + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{3} + 2 πn : \frac{2 π}{3} + 2 πn

2 \cdot \frac{π}{3} + 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

Solutions for the range

0 \leq x < 2 π

x = \frac{2 π}{3}

sin (x) = \frac{4}{cos ( x )}

tan^{3} (x) - 2 tan (x) = tan (x)

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

sin (x + 1 6^{\circ}) = cos (11 x + 2)

2 csc (θ) + 2 = 0

What is the general solution for tan(x/2)=sqrt(3),0<= x<2pi ?
The general solution for tan(x/2)=sqrt(3),0<= x<2pi is x=(2pi)/3