What is the general solution for tan(1/2 x)-sqrt(3)=0 ?

The general solution for tan(1/2 x)-sqrt(3)=0 is x=(2pi)/3+2pin

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tan(1/2 x)-sqrt(3)=0

tan (\frac{1}{2} x) - 3 = 0

x = \frac{2 π}{3} + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (\frac{1}{2} x) - 3 = 0

Move

3

to the right side

tan (\frac{1}{2} x) - 3 = 0

Add

3

to both sides

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

Simplify

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

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

General solutions for

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

\frac{1}{2} x = \frac{π}{3} + πn

Solve

\frac{1}{2} x = \frac{π}{3} + πn : x = \frac{2 π}{3} + 2 πn

\frac{1}{2} x = \frac{π}{3} + πn

Multiply both sides by

2

\frac{1}{2} x = \frac{π}{3} + πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{π}{3} + 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{π}{3} + 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

= \frac{1 \cdot 2}{2} x

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= 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

2 cos (2 θ) = 6 cos (θ) - 4

5 sin (θ) - 4 = 2 sin (θ) - 6

cos (2 θ) - cos (4 θ) = 0

tan (x) = 1.4

sin (θ) = 0.71

What is the general solution for tan(1/2 x)-sqrt(3)=0 ?
The general solution for tan(1/2 x)-sqrt(3)=0 is x=(2pi)/3+2pin