What is the general solution for csc(θ)=(-3)/(sqrt(5)) ?

The general solution for csc(θ)=(-3)/(sqrt(5)) is θ=-0.84106…+2pin,θ=pi+0.84106…+2pin

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csc(θ)=(-3)/(sqrt(5))

csc (θ) = \frac{- 3}{5}

θ = - 0.84106 \dots + 2 πn, θ = π + 0.84106 \dots + 2 πn

Radians

θ = - 0.84106 \dots + 6.28318 \dots n, θ = π + 0.84106 \dots + 6.28318 \dots n

Solution steps

csc (θ) = \frac{- 3}{5}

Simplify

\frac{- 3}{5} : - \frac{3 5}{5}

\frac{- 3}{5}

Apply the fraction rule:

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

= - \frac{3}{5}

Rationalize

- \frac{3}{5} : - \frac{3 5}{5}

- \frac{3}{5}

Multiply by the conjugate

\frac{5}{5}

= - \frac{3 5}{5 5}

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= - \frac{3 5}{5}

= - \frac{3 5}{5}

csc (θ) = - \frac{3 5}{5}

Apply trig inverse properties

csc (θ) = - \frac{3 5}{5}

General solutions for

csc (θ) = - \frac{3 5}{5}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

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

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

Show solutions in decimal form

θ = - 0.84106 \dots + 2 πn, θ = π + 0.84106 \dots + 2 πn

sec^{2} (x) + tan^{2} (x) = 5 tan (x)

sin (x) = \frac{9}{12}

- 2 cos (x) + 3 sin (x) = 2

cos (x + \frac{π}{6}) - 1 = cos (x - \frac{π}{6})

cos (2 x) cos (x) - sin (2 x) sin (x) = \frac{- 3}{2}

What is the general solution for csc(θ)=(-3)/(sqrt(5)) ?
The general solution for csc(θ)=(-3)/(sqrt(5)) is θ=-0.84106…+2pin,θ=pi+0.84106…+2pin