What is the general solution for csc^2(x)=5csc(x) ?

The general solution for csc^2(x)=5csc(x) is x=0.20135…+2pin,x=pi-0.20135…+2pin

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csc^2(x)=5csc(x)

Solution

csc^{2} (x) = 5 csc (x)

Solution

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

Degrees

x = 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 168.46304 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (x) = 5 csc (x)

Solve by substitution

csc^{2} (x) = 5 csc (x)

Let:

csc (x) = u

u^{2} = 5 u

u^{2} = 5 u : u = 5, u = 0

u^{2} = 5 u

Move

5 u

to the left side

u^{2} = 5 u

Subtract

5 u

from both sides

u^{2} - 5 u = 5 u - 5 u

Simplify

u^{2} - 5 u = 0

u^{2} - 5 u = 0

Solve with the quadratic formula

u^{2} - 5 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 5, c = 0

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot 0 = 5

(- 5)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

Apply radical rule:

assuming

a \geq 0

= 5

u_{1, 2} = \frac{- ( - 5 ) \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 5}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 5}{2 \cdot 1}

u = \frac{- ( - 5 ) + 5}{2 \cdot 1} : 5

\frac{- ( - 5 ) + 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 + 5}{2 \cdot 1}

Add the numbers:

5 + 5 = 10

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5

u = \frac{- ( - 5 ) - 5}{2 \cdot 1} : 0

\frac{- ( - 5 ) - 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 - 5}{2 \cdot 1}

Subtract the numbers:

5 - 5 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 5, u = 0

Substitute back

u = csc (x)

csc (x) = 5, csc (x) = 0

csc (x) = 5, csc (x) = 0

csc (x) = 5 : x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 5

Apply trig inverse properties

csc (x) = 5

General solutions for

csc (x) = 5

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

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

Combine all the solutions

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

Show solutions in decimal form

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc^2(x)=5csc(x) ?
The general solution for csc^2(x)=5csc(x) is x=0.20135…+2pin,x=pi-0.20135…+2pin