What is the general solution for csc^{(2)}(x)-cot(x)=2,0<,x<360 ?

Q: What is the general solution for csc^{(2)}(x)-cot(x)=2,0<,x<360 ?

The general solution for csc^{(2)}(x)-cot(x)=2,0<,x<360 is No Solution for x\in\mathbb{R}

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csc^{(2)}(x)-cot(x)=2,0<,x<360

Solution

csc^{(2)} (x) - cot (x) = 2, 0 <, x < 360

Solution

No Solution for x \in R

Solution steps

csc^{(2)} (x) - cot (x) = 2, 0, x < 360

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Group like terms

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

Add/Subtract the numbers:

- 2 + 1 = - 1

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

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

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

Solve by substitution

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

Let:

cot (x) = u

- 1 - u + u^{2} = 0

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

- 1 - u + u^{2} = 0

Write in the standard form

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

u^{2} - u - 1 = 0

Solve with the quadratic formula

u^{2} - u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 1, c = - 1

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

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

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

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

Apply rule

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 + 4

Add the numbers:

1 + 4 = 5

= 5

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{1 + 5}{2}

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{1 - 5}{2}

The solutions to the quadratic equation are:

u = \frac{1 + 5}{2}, u = \frac{1 - 5}{2}

Substitute back

u = cot (x)

cot (x) = \frac{1 + 5}{2}, cot (x) = \frac{1 - 5}{2}

cot (x) = \frac{1 + 5}{2}, cot (x) = \frac{1 - 5}{2}

cot (x) = \frac{1 + 5}{2}, 0 :

No Solution

cot (x) = \frac{1 + 5}{2}, 0

Apply trig inverse properties

cot (x) = \frac{1 + 5}{2}

General solutions for

cot (x) = \frac{1 + 5}{2}

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

x = arccot (\frac{1 + 5}{2}) + πn

x = arccot (\frac{1 + 5}{2}) + πn

Solutions for the range

0

No Solution

cot (x) = \frac{1 - 5}{2}, 0 :

No Solution

cot (x) = \frac{1 - 5}{2}, 0

Apply trig inverse properties

cot (x) = \frac{1 - 5}{2}

General solutions for

cot (x) = \frac{1 - 5}{2}

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

x = arccot (\frac{1 - 5}{2}) + πn

x = arccot (\frac{1 - 5}{2}) + πn

Solutions for the range

0

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for csc^{(2)}(x)-cot(x)=2,0<,x<360 ?
The general solution for csc^{(2)}(x)-cot(x)=2,0<,x<360 is No Solution for x\in\mathbb{R}