What is the general solution for -8csc^2(x)-4cot(x)=-12 ?

The general solution for -8csc^2(x)-4cot(x)=-12 is x=(3pi)/4+pin,x=1.10714…+pin

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-8csc^2(x)-4cot(x)=-12

Solution

- 8 csc^{2} (x) - 4 cot (x) = - 12

Solution

x = \frac{3 π}{4} + πn, x = 1.10714 \dots + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 8 csc^{2} (x) - 4 cot (x) = - 12

Subtract

- 12

from both sides

- 8 csc^{2} (x) - 4 cot (x) + 12 = 0

Rewrite using trig identities

12 - 4 cot (x) - 8 csc^{2} (x)

Use the Pythagorean identity:

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

= 12 - 4 cot (x) - 8 (1 + cot^{2} (x))

Simplify

12 - 4 cot (x) - 8 (1 + cot^{2} (x)) : - 8 cot^{2} (x) - 4 cot (x) + 4

12 - 4 cot (x) - 8 (1 + cot^{2} (x))

Expand

- 8 (1 + cot^{2} (x)) : - 8 - 8 cot^{2} (x)

- 8 (1 + cot^{2} (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - 8, b = 1, c = cot^{2} (x)

= - 8 \cdot 1 + (- 8) cot^{2} (x)

Apply minus-plus rules

+ (- a) = - a

= - 8 \cdot 1 - 8 cot^{2} (x)

Multiply the numbers:

8 \cdot 1 = 8

= - 8 - 8 cot^{2} (x)

= 12 - 4 cot (x) - 8 - 8 cot^{2} (x)

Simplify

12 - 4 cot (x) - 8 - 8 cot^{2} (x) : - 8 cot^{2} (x) - 4 cot (x) + 4

12 - 4 cot (x) - 8 - 8 cot^{2} (x)

Group like terms

= - 4 cot (x) - 8 cot^{2} (x) + 12 - 8

Add/Subtract the numbers:

12 - 8 = 4

= - 8 cot^{2} (x) - 4 cot (x) + 4

= - 8 cot^{2} (x) - 4 cot (x) + 4

= - 8 cot^{2} (x) - 4 cot (x) + 4

4 - 4 cot (x) - 8 cot^{2} (x) = 0

Solve by substitution

4 - 4 cot (x) - 8 cot^{2} (x) = 0

Let:

cot (x) = u

4 - 4 u - 8 u^{2} = 0

4 - 4 u - 8 u^{2} = 0 : u = - 1, u = \frac{1}{2}

4 - 4 u - 8 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 8, b = - 4, c = 4

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

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

(- 4)^{2} - 4 (- 8) \cdot 4 = 12

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 8 \cdot 4 = 128

= 4^{2} + 128

4^{2} = 16

= 16 + 128

Add the numbers:

16 + 128 = 144

= 144

Factor the number:

144 = 1 2^{2}

= 1 2^{2}

Apply radical rule:

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- ( - 4 ) \pm 12}{2 ( - 8 )}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 12}{2 ( - 8 )}, u_{2} = \frac{- ( - 4 ) - 12}{2 ( - 8 )}

u = \frac{- ( - 4 ) + 12}{2 ( - 8 )} : - 1

\frac{- ( - 4 ) + 12}{2 ( - 8 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{4 + 12}{- 2 \cdot 8}

Add the numbers:

4 + 12 = 16

= \frac{16}{- 2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{16}{- 16}

Apply the fraction rule:

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

= - \frac{16}{16}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 4 ) - 12}{2 ( - 8 )} : \frac{1}{2}

\frac{- ( - 4 ) - 12}{2 ( - 8 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{4 - 12}{- 2 \cdot 8}

Subtract the numbers:

4 - 12 = - 8

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

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 8}{- 16}

Apply the fraction rule:

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

= \frac{8}{16}

Cancel the common factor:

8

= \frac{1}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = cot (x)

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

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

cot (x) = - 1 : x = \frac{3 π}{4} + πn

cot (x) = - 1

General solutions for

cot (x) = - 1

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

Combine all the solutions

x = \frac{3 π}{4} + πn, x = arccot (\frac{1}{2}) + πn

Show solutions in decimal form

x = \frac{3 π}{4} + πn, x = 1.10714 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for -8csc^2(x)-4cot(x)=-12 ?
The general solution for -8csc^2(x)-4cot(x)=-12 is x=(3pi)/4+pin,x=1.10714…+pin