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Popular >

csc^2(θ)+2csc(θ)-24=0

Solution

csc^{2} (θ) + 2 csc (θ) - 24 = 0

Solution

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

+1

Degrees

θ = 14.47751 \dots^{\circ} + 36 0^{\circ} n, θ = 165.52248 \dots^{\circ} + 36 0^{\circ} n, θ = - 9.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 189.59406 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (θ) + 2 csc (θ) - 24 = 0

Solve by substitution

csc^{2} (θ) + 2 csc (θ) - 24 = 0

Let:

csc (θ) = u

u^{2} + 2 u - 24 = 0

u^{2} + 2 u - 24 = 0 : u = 4, u = - 6

u^{2} + 2 u - 24 = 0

Solve with the quadratic formula

u^{2} + 2 u - 24 = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = - 24

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 24 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 24 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 24) = 10

2^{2} - 4 \cdot 1 \cdot (- 24)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 24

Multiply the numbers:

4 \cdot 1 \cdot 24 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

Add the numbers:

4 + 96 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 2 + 10}{2 \cdot 1}, u_{2} = \frac{- 2 - 10}{2 \cdot 1}

u = \frac{- 2 + 10}{2 \cdot 1} : 4

\frac{- 2 + 10}{2 \cdot 1}

Add/Subtract the numbers:

- 2 + 10 = 8

= \frac{8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

u = \frac{- 2 - 10}{2 \cdot 1} : - 6

\frac{- 2 - 10}{2 \cdot 1}

Subtract the numbers:

- 2 - 10 = - 12

= \frac{- 12}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12}{2}

Apply the fraction rule:

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

= - \frac{12}{2}

Divide the numbers:

\frac{12}{2} = 6

= - 6

The solutions to the quadratic equation are:

u = 4, u = - 6

Substitute back

u = csc (θ)

csc (θ) = 4, csc (θ) = - 6

csc (θ) = 4, csc (θ) = - 6

csc (θ) = 4 : θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

csc (θ) = 4

Apply trig inverse properties

csc (θ) = 4

General solutions for

csc (θ) = 4

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

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

csc (θ) = - 6 : θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

csc (θ) = - 6

Apply trig inverse properties

csc (θ) = - 6

General solutions for

csc (θ) = - 6

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

θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

Combine all the solutions

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn, θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

Show solutions in decimal form

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

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