What is the general solution for 3csc^2(x)-2csc(x)-5=0 ?

The general solution for 3csc^2(x)-2csc(x)-5=0 is x=0.64350…+2pin,x=pi-0.64350…+2pin,x=(3pi)/2+2pin

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Popular >

3csc^2(x)-2csc(x)-5=0

Solution

3 csc^{2} (x) - 2 csc (x) - 5 = 0

Solution

x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

3 csc^{2} (x) - 2 csc (x) - 5 = 0

Solve by substitution

3 csc^{2} (x) - 2 csc (x) - 5 = 0

Let:

csc (x) = u

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

3 u^{2} - 2 u - 5 = 0 : u = \frac{5}{3}, u = - 1

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 3, b = - 2, c = - 5

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 3 ( - 5 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 3 ( - 5 )}{2 \cdot 3}

(- 2)^{2} - 4 \cdot 3 (- 5) = 8

(- 2)^{2} - 4 \cdot 3 (- 5)

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 3 \cdot 5

Apply exponent rule:

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

n

is even

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

= 2^{2} + 4 \cdot 3 \cdot 5

Multiply the numbers:

4 \cdot 3 \cdot 5 = 60

= 2^{2} + 60

2^{2} = 4

= 4 + 60

Add the numbers:

4 + 60 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 2 ) \pm 8}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 8}{2 \cdot 3}, u_{2} = \frac{- ( - 2 ) - 8}{2 \cdot 3}

u = \frac{- ( - 2 ) + 8}{2 \cdot 3} : \frac{5}{3}

\frac{- ( - 2 ) + 8}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{2 + 8}{2 \cdot 3}

Add the numbers:

2 + 8 = 10

= \frac{10}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{10}{6}

Cancel the common factor:

2

= \frac{5}{3}

u = \frac{- ( - 2 ) - 8}{2 \cdot 3} : - 1

\frac{- ( - 2 ) - 8}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{2 - 8}{2 \cdot 3}

Subtract the numbers:

2 - 8 = - 6

= \frac{- 6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6}{6}

Apply the fraction rule:

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

= - \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = \frac{5}{3}, u = - 1

Substitute back

u = csc (x)

csc (x) = \frac{5}{3}, csc (x) = - 1

csc (x) = \frac{5}{3}, csc (x) = - 1

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

csc (x) = \frac{5}{3}

Apply trig inverse properties

csc (x) = \frac{5}{3}

General solutions for

csc (x) = \frac{5}{3}

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

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

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

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

General solutions for

csc (x) = - 1

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3csc^2(x)-2csc(x)-5=0 ?
The general solution for 3csc^2(x)-2csc(x)-5=0 is x=0.64350…+2pin,x=pi-0.64350…+2pin,x=(3pi)/2+2pin