What is the general solution for csc^2(x)=5csc(x)+6 ?

The general solution for csc^2(x)=5csc(x)+6 is x=0.16744…+2pin,x=pi-0.16744…+2pin,x=(3pi)/2+2pin

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csc^2(x)=5csc(x)+6

Solution

csc^{2} (x) = 5 csc (x) + 6

Solution

x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9.59406 \dots^{\circ} + 36 0^{\circ} n, x = 170.40593 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (x) = 5 csc (x) + 6

Solve by substitution

csc^{2} (x) = 5 csc (x) + 6

Let:

csc (x) = u

u^{2} = 5 u + 6

u^{2} = 5 u + 6 : u = 6, u = - 1

u^{2} = 5 u + 6

Move

6

to the left side

u^{2} = 5 u + 6

Subtract

6

from both sides

u^{2} - 6 = 5 u + 6 - 6

Simplify

u^{2} - 6 = 5 u

u^{2} - 6 = 5 u

Move

5 u

to the left side

u^{2} - 6 = 5 u

Subtract

5 u

from both sides

u^{2} - 6 - 5 u = 5 u - 5 u

Simplify

u^{2} - 6 - 5 u = 0

u^{2} - 6 - 5 u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - 5 u - 6 = 0

Solve with the quadratic formula

u^{2} - 5 u - 6 = 0

Quadratic Equation Formula:

For

a = 1, b = - 5, c = - 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot (- 6) = 7

(- 5)^{2} - 4 \cdot 1 \cdot (- 6)

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 1 \cdot 6

Apply exponent rule:

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

n

is even

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

= 5^{2} + 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 7}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 7}{2 \cdot 1}

u = \frac{- ( - 5 ) + 7}{2 \cdot 1} : 6

\frac{- ( - 5 ) + 7}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 + 7}{2 \cdot 1}

Add the numbers:

5 + 7 = 12

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6

u = \frac{- ( - 5 ) - 7}{2 \cdot 1} : - 1

\frac{- ( - 5 ) - 7}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{5 - 7}{2 \cdot 1}

Subtract the numbers:

5 - 7 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 6, u = - 1

Substitute back

u = csc (x)

csc (x) = 6, csc (x) = - 1

csc (x) = 6, csc (x) = - 1

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

csc (x) = 6

Apply trig inverse properties

csc (x) = 6

General solutions for

csc (x) = 6

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

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

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

Show solutions in decimal form

x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Popular Examples

tan(θ)=1770

tan (θ) = 1770

2sin^2(x)-5/4 =-cos^2(x)

2 sin^{2} (x) - \frac{5}{4} = - cos^{2} (x)

2sqrt(3)sin(2x)=sqrt(3)

23 sin (2 x) = 3

(sin^2(x)-tan^2(x))/(csc(x))=sin(x)

\frac{sin ^{2} ( x ) - tan ^{2} ( x )}{csc ( x )} = sin (x)

sin(x)=(2*490*10^{-9})/(151*10^{-6)}

sin (x) = \frac{2 \cdot 490 \cdot 1 0 ^{- 9}}{151 \cdot 1 0 ^{- 6}}

Frequently Asked Questions (FAQ)

What is the general solution for csc^2(x)=5csc(x)+6 ?
The general solution for csc^2(x)=5csc(x)+6 is x=0.16744…+2pin,x=pi-0.16744…+2pin,x=(3pi)/2+2pin