What is the general solution for csc(θ/3)=sqrt(6) ?

The general solution for csc(θ/3)=sqrt(6) is θ=3*0.42053…+6pin,θ=3pi-3*0.42053…+6pin

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csc(θ/3)=sqrt(6)

csc (\frac{θ}{3}) = 6

θ = 3 \cdot 0.42053 \dots + 6 πn, θ = 3 π - 3 \cdot 0.42053 \dots + 6 πn

Degrees

θ = 72.28452 \dots^{\circ} + 108 0^{\circ} n, θ = 467.71547 \dots^{\circ} + 108 0^{\circ} n

Solution steps

csc (\frac{θ}{3}) = 6

Apply trig inverse properties

csc (\frac{θ}{3}) = 6

General solutions for

csc (\frac{θ}{3}) = 6

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

\frac{θ}{3} = arccsc (6) + 2 πn, \frac{θ}{3} = π - arccsc (6) + 2 πn

\frac{θ}{3} = arccsc (6) + 2 πn, \frac{θ}{3} = π - arccsc (6) + 2 πn

Solve

\frac{θ}{3} = arccsc (6) + 2 πn : θ = 3 arccsc (6) + 6 πn

\frac{θ}{3} = arccsc (6) + 2 πn

Multiply both sides by

3

\frac{θ}{3} = arccsc (6) + 2 πn

Multiply both sides by

3

\frac{3 θ}{3} = 3 arccsc (6) + 3 \cdot 2 πn

Simplify

\frac{3 θ}{3} = 3 arccsc (6) + 3 \cdot 2 πn

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

3 arccsc (6) + 3 \cdot 2 πn : 3 arccsc (6) + 6 πn

3 arccsc (6) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arccsc (6) + 6 πn

θ = 3 arccsc (6) + 6 πn

θ = 3 arccsc (6) + 6 πn

θ = 3 arccsc (6) + 6 πn

Solve

\frac{θ}{3} = π - arccsc (6) + 2 πn : θ = 3 π - 3 arccsc (6) + 6 πn

\frac{θ}{3} = π - arccsc (6) + 2 πn

Multiply both sides by

3

\frac{θ}{3} = π - arccsc (6) + 2 πn

Multiply both sides by

3

\frac{3 θ}{3} = 3 π - 3 arccsc (6) + 3 \cdot 2 πn

Simplify

\frac{3 θ}{3} = 3 π - 3 arccsc (6) + 3 \cdot 2 πn

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

3 π - 3 arccsc (6) + 3 \cdot 2 πn : 3 π - 3 arccsc (6) + 6 πn

3 π - 3 arccsc (6) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 π - 3 arccsc (6) + 6 πn

θ = 3 π - 3 arccsc (6) + 6 πn

θ = 3 π - 3 arccsc (6) + 6 πn

θ = 3 π - 3 arccsc (6) + 6 πn

θ = 3 arccsc (6) + 6 πn, θ = 3 π - 3 arccsc (6) + 6 πn

Show solutions in decimal form

θ = 3 \cdot 0.42053 \dots + 6 πn, θ = 3 π - 3 \cdot 0.42053 \dots + 6 πn

arcsin (x) = 0

tan (x) = 1.5

tan (3 x) = - 3

1 = cos (x)

(cot (θ) - 1) (2 sin (θ) + 1) = 0

What is the general solution for csc(θ/3)=sqrt(6) ?
The general solution for csc(θ/3)=sqrt(6) is θ=3*0.42053…+6pin,θ=3pi-3*0.42053…+6pin