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

The general solution for tan(θ)=(-1)/(sqrt(3)) is θ=(5pi)/6+pin

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tan(θ)=(-1)/(sqrt(3))

tan (θ) = \frac{- 1}{3}

θ = \frac{5 π}{6} + πn

Solution steps

tan (θ) = \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 (θ) = - \frac{3}{3}

General solutions for

tan (θ) = - \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}

θ = \frac{5 π}{6} + πn

θ = \frac{5 π}{6} + πn

cos (x) = - \frac{1}{2}, 0 \leq x \leq 2 π

cot^{2} (θ) + cot (θ) - 20 = 0

cos (2 x) + 2 cos (x) + 1 = 0

sin^{2} (x) + 1 = cos^{2} (x)

arcsin (x) = \frac{π}{2}

What is the general solution for tan(θ)=(-1)/(sqrt(3)) ?
The general solution for tan(θ)=(-1)/(sqrt(3)) is θ=(5pi)/6+pin