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

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

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tan(θ/2)=sqrt(3)

tan (\frac{θ}{2}) = 3

θ = \frac{2 π}{3} + 2 πn

Degrees

θ = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (\frac{θ}{2}) = 3

General solutions for

tan (\frac{θ}{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{θ}{2} = \frac{π}{3} + πn

\frac{θ}{2} = \frac{π}{3} + πn

Solve

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

\frac{θ}{2} = \frac{π}{3} + πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{3} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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

θ = \frac{2 π}{3} + 2 πn

θ = \frac{2 π}{3} + 2 πn

θ = \frac{2 π}{3} + 2 πn

θ = \frac{2 π}{3} + 2 πn

tan (x) = \frac{1}{4}

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

3 tan (2 x) + 1 = 0

sin (x) = \frac{2}{π}

sin (2 x) cos (x) + cos (2 x) sin (x) = \frac{2}{2}

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