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

The general solution for csc^2(x)+2cot(x)=1 is x= pi/2+pin,x=2.67794…+pin

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

Solution

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

Solution

x = \frac{π}{2} + πn, x = 2.67794 \dots + πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n, x = 153.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Solve by substitution

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

Let:

cot (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 = cot (x)

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

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

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

General solutions for

cot (x) = 0

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

cot (x) = - 2 : x = arccot (- 2) + πn

cot (x) = - 2

Apply trig inverse properties

cot (x) = - 2

General solutions for

cot (x) = - 2

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2) + πn

x = arccot (- 2) + πn

Combine all the solutions

x = \frac{π}{2} + πn, x = arccot (- 2) + πn

Show solutions in decimal form

x = \frac{π}{2} + πn, x = 2.67794 \dots + πn

Popular Examples

2cos(x)+sin(x)=2

2 cos (x) + sin (x) = 2

sin(θ)tan(θ)-sin(θ)=0

sin (θ) tan (θ) - sin (θ) = 0

6cos^2(x)+5sin(x)=7

6 cos^{2} (x) + 5 sin (x) = 7

sin(θ)=0.9848,0<= θ<360

sin (θ) = 0.9848, 0^{\circ} \leq θ < 36 0^{\circ}

sin^2(x)=((7m-9))/9

sin^{2} (x) = \frac{( 7 m - 9 )}{9}

Frequently Asked Questions (FAQ)

What is the general solution for csc^2(x)+2cot(x)=1 ?
The general solution for csc^2(x)+2cot(x)=1 is x= pi/2+pin,x=2.67794…+pin