What is the general solution for -6csc^2(x)+10cot(x)=-10 ?

The general solution for -6csc^2(x)+10cot(x)=-10 is x=1.89254…+pin,x=0.46364…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

-6csc^2(x)+10cot(x)=-10

Solution

- 6 csc^{2} (x) + 10 cot (x) = - 10

Solution

x = 1.89254 \dots + πn, x = 0.46364 \dots + πn

Degrees

x = 108.43494 \dots^{\circ} + 18 0^{\circ} n, x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 6 csc^{2} (x) + 10 cot (x) = - 10

Subtract

- 10

from both sides

- 6 csc^{2} (x) + 10 cot (x) + 10 = 0

Rewrite using trig identities

10 + 10 cot (x) - 6 csc^{2} (x)

Use the Pythagorean identity:

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

= 10 + 10 cot (x) - 6 (1 + cot^{2} (x))

Simplify

10 + 10 cot (x) - 6 (1 + cot^{2} (x)) : 10 cot (x) - 6 cot^{2} (x) + 4

10 + 10 cot (x) - 6 (1 + cot^{2} (x))

Expand

- 6 (1 + cot^{2} (x)) : - 6 - 6 cot^{2} (x)

- 6 (1 + cot^{2} (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - 6, b = 1, c = cot^{2} (x)

= - 6 \cdot 1 + (- 6) cot^{2} (x)

Apply minus-plus rules

+ (- a) = - a

= - 6 \cdot 1 - 6 cot^{2} (x)

Multiply the numbers:

6 \cdot 1 = 6

= - 6 - 6 cot^{2} (x)

= 10 + 10 cot (x) - 6 - 6 cot^{2} (x)

Simplify

10 + 10 cot (x) - 6 - 6 cot^{2} (x) : 10 cot (x) - 6 cot^{2} (x) + 4

10 + 10 cot (x) - 6 - 6 cot^{2} (x)

Group like terms

= 10 cot (x) - 6 cot^{2} (x) + 10 - 6

Add/Subtract the numbers:

10 - 6 = 4

= 10 cot (x) - 6 cot^{2} (x) + 4

= 10 cot (x) - 6 cot^{2} (x) + 4

= 10 cot (x) - 6 cot^{2} (x) + 4

4 + 10 cot (x) - 6 cot^{2} (x) = 0

Solve by substitution

4 + 10 cot (x) - 6 cot^{2} (x) = 0

Let:

cot (x) = u

4 + 10 u - 6 u^{2} = 0

4 + 10 u - 6 u^{2} = 0 : u = - \frac{1}{3}, u = 2

4 + 10 u - 6 u^{2} = 0

Write in the standard form

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

- 6 u^{2} + 10 u + 4 = 0

Solve with the quadratic formula

- 6 u^{2} + 10 u + 4 = 0

Quadratic Equation Formula:

For

a = - 6, b = 10, c = 4

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 6 ) \cdot 4}{2 ( - 6 )}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 6 ) \cdot 4}{2 ( - 6 )}

1 0^{2} - 4 (- 6) \cdot 4 = 14

1 0^{2} - 4 (- 6) \cdot 4

Apply rule

- (- a) = a

= 1 0^{2} + 4 \cdot 6 \cdot 4

Multiply the numbers:

4 \cdot 6 \cdot 4 = 96

= 1 0^{2} + 96

1 0^{2} = 100

= 100 + 96

Add the numbers:

100 + 96 = 196

= 196

Factor the number:

196 = 1 4^{2}

= 1 4^{2}

Apply radical rule:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- 10 \pm 14}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- 10 + 14}{2 ( - 6 )}, u_{2} = \frac{- 10 - 14}{2 ( - 6 )}

u = \frac{- 10 + 14}{2 ( - 6 )} : - \frac{1}{3}

\frac{- 10 + 14}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 10 + 14}{- 2 \cdot 6}

Add/Subtract the numbers:

- 10 + 14 = 4

= \frac{4}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4}{- 12}

Apply the fraction rule:

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

= - \frac{4}{12}

Cancel the common factor:

4

= - \frac{1}{3}

u = \frac{- 10 - 14}{2 ( - 6 )} : 2

\frac{- 10 - 14}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 10 - 14}{- 2 \cdot 6}

Subtract the numbers:

- 10 - 14 = - 24

= \frac{- 24}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 24}{- 12}

Apply the fraction rule:

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

= \frac{24}{12}

Divide the numbers:

\frac{24}{12} = 2

= 2

The solutions to the quadratic equation are:

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

Substitute back

u = cot (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (- \frac{1}{3}) + πn

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

Show solutions in decimal form

x = 1.89254 \dots + πn, x = 0.46364 \dots + πn

Popular Examples

solvefor x,cos(x)-cos(y)=0 solve for

x, cos (x) - cos (y) = 0

-3tan^2(θ)+2tan(θ)=6tan(θ)+1

- 3 tan^{2} (θ) + 2 tan (θ) = 6 tan (θ) + 1

sin(x)+cos(x)=-3

sin (x) + cos (x) = - 3

sin(x)= 1/2 ,cos(x)=-(sqrt(3))/2

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

cos(x)=(2sqrt(5))/5

cos (x) = \frac{2 5}{5}

Frequently Asked Questions (FAQ)

What is the general solution for -6csc^2(x)+10cot(x)=-10 ?
The general solution for -6csc^2(x)+10cot(x)=-10 is x=1.89254…+pin,x=0.46364…+pin