What is the general solution for csc(θ+pi)=-(2sqrt(3))/3 ?

The general solution for csc(θ+pi)=-(2sqrt(3))/3 is θ=(4pi)/3+2pin-pi,θ=(5pi)/3+2pin-pi

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csc(θ+pi)=-(2sqrt(3))/3

csc (θ + π) = - \frac{2 3}{3}

θ = \frac{4 π}{3} + 2 πn - π, θ = \frac{5 π}{3} + 2 πn - π

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

csc (θ + π) = - \frac{2 3}{3}

General solutions for

csc (θ + π) = - \frac{2 3}{3}

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

θ + π = \frac{4 π}{3} + 2 πn, θ + π = \frac{5 π}{3} + 2 πn

θ + π = \frac{4 π}{3} + 2 πn, θ + π = \frac{5 π}{3} + 2 πn

Solve

θ + π = \frac{4 π}{3} + 2 πn : θ = \frac{4 π}{3} + 2 πn - π

θ + π = \frac{4 π}{3} + 2 πn

Move

π

to the right side

θ + π = \frac{4 π}{3} + 2 πn

Subtract

π

from both sides

θ + π - π = \frac{4 π}{3} + 2 πn - π

Simplify

θ = \frac{4 π}{3} + 2 πn - π

θ = \frac{4 π}{3} + 2 πn - π