What is the general solution for csc^2(x)-3csc(x)-2=0 ?

The general solution for csc^2(x)-3csc(x)-2=0 is x=0.28460…+2pin,x=pi-0.28460…+2pin

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csc^2(x)-3csc(x)-2=0

Solution

csc^{2} (x) - 3 csc (x) - 2 = 0

Solution

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

Degrees

x = 16.30654 \dots^{\circ} + 36 0^{\circ} n, x = 163.69345 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (x) - 3 csc (x) - 2 = 0

Solve by substitution

csc^{2} (x) - 3 csc (x) - 2 = 0

Let:

csc (x) = u

u^{2} - 3 u - 2 = 0

u^{2} - 3 u - 2 = 0 : u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

u^{2} - 3 u - 2 = 0

Solve with the quadratic formula

u^{2} - 3 u - 2 = 0

Quadratic Equation Formula:

For

a = 1, b = - 3, c = - 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot ( - 2 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot ( - 2 )}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot (- 2) = 17

(- 3)^{2} - 4 \cdot 1 \cdot (- 2)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 2

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

Add the numbers:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- ( - 3 ) \pm 17}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 17}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 17}{2 \cdot 1}

u = \frac{- ( - 3 ) + 17}{2 \cdot 1} : \frac{3 + 17}{2}

\frac{- ( - 3 ) + 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 17}{2}

u = \frac{- ( - 3 ) - 17}{2 \cdot 1} : \frac{3 - 17}{2}

\frac{- ( - 3 ) - 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 17}{2}

The solutions to the quadratic equation are:

u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

Substitute back

u = csc (x)

csc (x) = \frac{3 + 17}{2}, csc (x) = \frac{3 - 17}{2}

csc (x) = \frac{3 + 17}{2}, csc (x) = \frac{3 - 17}{2}

csc (x) = \frac{3 + 17}{2} : x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

csc (x) = \frac{3 + 17}{2}

Apply trig inverse properties

csc (x) = \frac{3 + 17}{2}

General solutions for

csc (x) = \frac{3 + 17}{2}

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

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

csc (x) = \frac{3 - 17}{2} :

No Solution

csc (x) = \frac{3 - 17}{2}

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

Show solutions in decimal form

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc^2(x)-3csc(x)-2=0 ?
The general solution for csc^2(x)-3csc(x)-2=0 is x=0.28460…+2pin,x=pi-0.28460…+2pin