What is the general solution for 2cot^2(x)+2csc^2(x)=1+4csc(x) ?

The general solution for 2cot^2(x)+2csc^2(x)=1+4csc(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin

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2cot^2(x)+2csc^2(x)=1+4csc(x)

Solution

2 cot^{2} (x) + 2 csc^{2} (x) = 1 + 4 csc (x)

Solution

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Degrees

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cot^{2} (x) + 2 csc^{2} (x) = 1 + 4 csc (x)

Subtract

1 + 4 csc (x)

from both sides

2 cot^{2} (x) + 2 csc^{2} (x) - 1 - 4 csc (x) = 0

Rewrite using trig identities

- 1 + 2 cot^{2} (x) + 2 csc^{2} (x) - 4 csc (x)

Use the Pythagorean identity:

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

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

= - 1 + 2 (csc^{2} (x) - 1) + 2 csc^{2} (x) - 4 csc (x)

Simplify

- 1 + 2 (csc^{2} (x) - 1) + 2 csc^{2} (x) - 4 csc (x) : 4 csc^{2} (x) - 4 csc (x) - 3

- 1 + 2 (csc^{2} (x) - 1) + 2 csc^{2} (x) - 4 csc (x)

Expand

2 (csc^{2} (x) - 1) : 2 csc^{2} (x) - 2

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = csc^{2} (x), c = 1

= 2 csc^{2} (x) - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

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

= - 1 + 2 csc^{2} (x) - 2 + 2 csc^{2} (x) - 4 csc (x)

Simplify

- 1 + 2 csc^{2} (x) - 2 + 2 csc^{2} (x) - 4 csc (x) : 4 csc^{2} (x) - 4 csc (x) - 3

- 1 + 2 csc^{2} (x) - 2 + 2 csc^{2} (x) - 4 csc (x)

Group like terms

= 2 csc^{2} (x) + 2 csc^{2} (x) - 4 csc (x) - 1 - 2

Add similar elements:

2 csc^{2} (x) + 2 csc^{2} (x) = 4 csc^{2} (x)

= 4 csc^{2} (x) - 4 csc (x) - 1 - 2

Subtract the numbers:

- 1 - 2 = - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

= 4 csc^{2} (x) - 4 csc (x) - 3

- 3 - 4 csc (x) + 4 csc^{2} (x) = 0

Solve by substitution

- 3 - 4 csc (x) + 4 csc^{2} (x) = 0

Let:

csc (x) = u

- 3 - 4 u + 4 u^{2} = 0

- 3 - 4 u + 4 u^{2} = 0 : u = \frac{3}{2}, u = - \frac{1}{2}

- 3 - 4 u + 4 u^{2} = 0

Write in the standard form

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

4 u^{2} - 4 u - 3 = 0

Solve with the quadratic formula

4 u^{2} - 4 u - 3 = 0

Quadratic Equation Formula:

For

a = 4, b = - 4, c = - 3

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

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

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

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 4}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 4}

u = \frac{- ( - 4 ) + 8}{2 \cdot 4} : \frac{3}{2}

\frac{- ( - 4 ) + 8}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{4 + 8}{2 \cdot 4}

Add the numbers:

4 + 8 = 12

= \frac{12}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{12}{8}

Cancel the common factor:

4

= \frac{3}{2}

u = \frac{- ( - 4 ) - 8}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( - 4 ) - 8}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{4 - 8}{2 \cdot 4}

Subtract the numbers:

4 - 8 = - 4

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - \frac{1}{2}

Substitute back

u = csc (x)

csc (x) = \frac{3}{2}, csc (x) = - \frac{1}{2}

csc (x) = \frac{3}{2}, csc (x) = - \frac{1}{2}

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

csc (x) = \frac{3}{2}

Apply trig inverse properties

csc (x) = \frac{3}{2}

General solutions for

csc (x) = \frac{3}{2}

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

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

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

csc (x) = - \frac{1}{2} :

No Solution

csc (x) = - \frac{1}{2}

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

No Solution

Combine all the solutions

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

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2cot^2(x)+2csc^2(x)=1+4csc(x) ?
The general solution for 2cot^2(x)+2csc^2(x)=1+4csc(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin