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

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

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tan(x/2)=-(sqrt(3))/3

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

x = \frac{5 π}{3} + 2 πn

Degrees

x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

General solutions for

tan (\frac{x}{2}) = - \frac{3}{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{5 π}{6} + πn

\frac{x}{2} = \frac{5 π}{6} + πn

Solve

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

\frac{x}{2} = \frac{5 π}{6} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{5 π}{6} + 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

= \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

x = \frac{5 π}{3} + 2 πn

sec (3 x) - 2 = 0

cot (x) - 3 = 0

tan (x) = cot (x)

sin^{2} (θ) - 2 sin (θ) = 0

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

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