What is the general solution for tan(2x-pi/6)=(-1)/(sqrt(3)) ?

The general solution for tan(2x-pi/6)=(-1)/(sqrt(3)) is x=(pin)/2+pi/2

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tan(2x-pi/6)=(-1)/(sqrt(3))

Solution

tan (2 x - \frac{π}{6}) = \frac{- 1}{3}

Solution

x = \frac{πn}{2} + \frac{π}{2}

Radians

x = \frac{π}{2} + \frac{π}{2} n

Solution steps

tan (2 x - \frac{π}{6}) = \frac{- 1}{3}

Simplify

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

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \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}

= - \frac{3}{3}

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

General solutions for

tan (2 x - \frac{π}{6}) = - \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}

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

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

Solve

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

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

Move

\frac{π}{6}

to the right side

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

Add

\frac{π}{6}

to both sides

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

Simplify

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

Simplify

2 x - \frac{π}{6} + \frac{π}{6} : 2 x

2 x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= 2 x

Simplify

\frac{5 π}{6} + πn + \frac{π}{6} : πn + π

\frac{5 π}{6} + πn + \frac{π}{6}

Group like terms

= πn + \frac{π}{6} + \frac{5 π}{6}

Combine the fractions

\frac{π}{6} + \frac{5 π}{6} : π

Apply rule

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

= \frac{π + 5 π}{6}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{6}

Divide the numbers:

\frac{6}{6} = 1

= π

= πn + π

2 x = πn + π

2 x = πn + π

2 x = πn + π

Divide both sides by

2

2 x = πn + π

Divide both sides by

2

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

Simplify

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(2x-pi/6)=(-1)/(sqrt(3)) ?
The general solution for tan(2x-pi/6)=(-1)/(sqrt(3)) is x=(pin)/2+pi/2