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

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

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-sqrt(3)csc(θ)=2

- 3 csc (θ) = 2

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

Degrees

θ = 24 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

- 3 csc (θ) = 2

Divide both sides by

- 3

- 3 csc (θ) = 2

Divide both sides by

- 3

\frac{- 3 csc ( θ )}{- 3} = \frac{2}{- 3}

Simplify

\frac{- 3 csc ( θ )}{- 3} = \frac{2}{- 3}

Simplify

\frac{- 3 csc ( θ )}{- 3} : csc (θ)

\frac{- 3 csc ( θ )}{- 3}

Apply the fraction rule:

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

= \frac{3 csc ( θ )}{3}

Cancel the common factor:

3

= csc (θ)

Simplify

\frac{2}{- 3} : - \frac{2 3}{3}

\frac{2}{- 3}

Apply the fraction rule:

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

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

csc (θ) = - \frac{2 3}{3}

csc (θ) = - \frac{2 3}{3}

csc (θ) = - \frac{2 3}{3}

General solutions for

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

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

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

- 2 + 2 cos (x) + 4 cos^{2} (x) = 0

sin (A) = \frac{6.347}{12.7}

sec (x) = 3 cos (x) - tan (x)

arctan (x^{2} - 2 x) = 0

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