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

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

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

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

x = \frac{π}{3} + 2 πn

Solution steps

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

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

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{π}{6} + πn

\frac{x}{2} = \frac{π}{6} + πn

Solve

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

\frac{x}{2} = \frac{π}{6} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{π}{6} + 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

x = \frac{π}{3} + 2 πn

6 cos (x) = 5

6 cos (2 x) = 6 cos^{2} (x) - 5

1 - cos (x) = - sin (x)

tan (x) = 1, - 2 π \leq x \leq 2 π

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

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