What is the general solution for cot^2(x)+2csc(x)=-1 ?

The general solution for cot^2(x)+2csc(x)=-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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cot^2(x)+2csc(x)=-1

Solution

cot^{2} (x) + 2 csc (x) = - 1

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

cot^{2} (x) + 2 csc (x) = - 1

Subtract

- 1

from both sides

cot^{2} (x) + 2 csc (x) + 1 = 0

Rewrite using trig identities

1 + cot^{2} (x) + 2 csc (x)

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

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

csc^{2} (x) + 2 csc (x) = 0

Solve by substitution

csc^{2} (x) + 2 csc (x) = 0

Let:

csc (x) = u

u^{2} + 2 u = 0

u^{2} + 2 u = 0 : u = 0, u = - 2

u^{2} + 2 u = 0

Solve with the quadratic formula

u^{2} + 2 u = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = 0

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

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

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

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

Apply rule

0 \cdot a = 0

= 2^{2} - 0

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

= 2^{2}

Apply radical rule:

assuming

a \geq 0

= 2

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

Separate the solutions

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

u = \frac{- 2 + 2}{2 \cdot 1} : 0

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

Add/Subtract the numbers:

- 2 + 2 = 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

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

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

Subtract the numbers:

- 2 - 2 = - 4

= \frac{- 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4}{2}

Apply the fraction rule:

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

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

The solutions to the quadratic equation are:

u = 0, u = - 2

Substitute back

u = csc (x)

csc (x) = 0, csc (x) = - 2

csc (x) = 0, csc (x) = - 2

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - 2

General solutions for

csc (x) = - 2

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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

3cos(θ)-1=0

3 cos (θ) - 1 = 0

sqrt(2)cos^2(θ)-cos(θ)=0

2 cos^{2} (θ) - cos (θ) = 0

cos^2(x)+cos(x)-2=0

cos^{2} (x) + cos (x) - 2 = 0

cos(x)=sqrt(1/2)

cos (x) = \frac{1}{2}

cos(x)=-1,0<= x<= 2pi

cos (x) = - 1, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

What is the general solution for cot^2(x)+2csc(x)=-1 ?
The general solution for cot^2(x)+2csc(x)=-1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin