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

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

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cot(x/3)=(-1)/(sqrt(3))

cot (\frac{x}{3}) = \frac{- 1}{3}

x = 2 π + 3 πn

Solution steps

cot (\frac{x}{3}) = \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}

cot (\frac{x}{3}) = - \frac{3}{3}

General solutions for

cot (\frac{x}{3}) = - \frac{3}{3}

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

\frac{x}{3} = \frac{2 π}{3} + πn

\frac{x}{3} = \frac{2 π}{3} + πn

Solve

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

\frac{x}{3} = \frac{2 π}{3} + πn

Multiply both sides by

3

\frac{x}{3} = \frac{2 π}{3} + πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

3 \cdot \frac{2 π}{3} + 3 πn

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

= 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

cos^{2} (θ) = sin^{2} (θ) + 1, 0^{\circ} \leq θ \leq 36 0^{\circ}

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

tan (x) = 25762

sin (8 x) = sin (6 x)

\frac{1}{sin ( x )} = 2

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