What is the general solution for 2csc^2(x)=5-5cot(x) ?

The general solution for 2csc^2(x)=5-5cot(x) is x=1.10714…+pin,x=2.81984…+pin

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2csc^2(x)=5-5cot(x)

Solution

2 csc^{2} (x) = 5 - 5 cot (x)

Solution

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = 161.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 csc^{2} (x) = 5 - 5 cot (x)

Subtract

5 - 5 cot (x)

from both sides

2 csc^{2} (x) - 5 + 5 cot (x) = 0

Rewrite using trig identities

- 5 + 2 csc^{2} (x) + 5 cot (x)

Use the Pythagorean identity:

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

= - 5 + 2 (1 + cot^{2} (x)) + 5 cot (x)

Simplify

- 5 + 2 (1 + cot^{2} (x)) + 5 cot (x) : 2 cot^{2} (x) + 5 cot (x) - 3

- 5 + 2 (1 + cot^{2} (x)) + 5 cot (x)

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

= 2 \cdot 1 + 2 cot^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

= - 5 + 2 + 2 cot^{2} (x) + 5 cot (x)

Add/Subtract the numbers:

- 5 + 2 = - 3

= 2 cot^{2} (x) + 5 cot (x) - 3

= 2 cot^{2} (x) + 5 cot (x) - 3

- 3 + 2 cot^{2} (x) + 5 cot (x) = 0

Solve by substitution

- 3 + 2 cot^{2} (x) + 5 cot (x) = 0

Let:

cot (x) = u

- 3 + 2 u^{2} + 5 u = 0

- 3 + 2 u^{2} + 5 u = 0 : u = \frac{1}{2}, u = - 3

- 3 + 2 u^{2} + 5 u = 0

Write in the standard form

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

2 u^{2} + 5 u - 3 = 0

Solve with the quadratic formula

2 u^{2} + 5 u - 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 5, c = - 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

5^{2} - 4 \cdot 2 (- 3) = 7

5^{2} - 4 \cdot 2 (- 3)

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 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 2}

Separate the solutions

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

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

\frac{- 5 + 7}{2 \cdot 2}

Add/Subtract the numbers:

- 5 + 7 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- 5 - 7}{2 \cdot 2} : - 3

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

Subtract the numbers:

- 5 - 7 = - 12

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 12}{4}

Apply the fraction rule:

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

= - \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= - 3

The solutions to the quadratic equation are:

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

Substitute back

u = cot (x)

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

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

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

cot (x) = - 3 : x = arccot (- 3) + πn

cot (x) = - 3

Apply trig inverse properties

cot (x) = - 3

General solutions for

cot (x) = - 3

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

x = arccot (- 3) + πn

x = arccot (- 3) + πn

Combine all the solutions

x = arccot (\frac{1}{2}) + πn, x = arccot (- 3) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2csc^2(x)=5-5cot(x) ?
The general solution for 2csc^2(x)=5-5cot(x) is x=1.10714…+pin,x=2.81984…+pin