What is the general solution for 3cot^2(x)-14csc(x)-2=0 ?

The general solution for 3cot^2(x)-14csc(x)-2=0 is x=0.20135…+2pin,x=pi-0.20135…+2pin

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3cot^2(x)-14csc(x)-2=0

Solution

3 cot^{2} (x) - 14 csc (x) - 2 = 0

Solution

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

Degrees

x = 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 168.46304 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cot^{2} (x) - 14 csc (x) - 2 = 0

Rewrite using trig identities

- 2 - 14 csc (x) + 3 cot^{2} (x)

Use the Pythagorean identity:

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

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

= - 2 - 14 csc (x) + 3 (csc^{2} (x) - 1)

Simplify

- 2 - 14 csc (x) + 3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 14 csc (x) - 5

- 2 - 14 csc (x) + 3 (csc^{2} (x) - 1)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

= 3 csc^{2} (x) - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

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

= - 2 - 14 csc (x) + 3 csc^{2} (x) - 3

Simplify

- 2 - 14 csc (x) + 3 csc^{2} (x) - 3 : 3 csc^{2} (x) - 14 csc (x) - 5

- 2 - 14 csc (x) + 3 csc^{2} (x) - 3

Group like terms

= - 14 csc (x) + 3 csc^{2} (x) - 2 - 3

Subtract the numbers:

- 2 - 3 = - 5

= 3 csc^{2} (x) - 14 csc (x) - 5

= 3 csc^{2} (x) - 14 csc (x) - 5

= 3 csc^{2} (x) - 14 csc (x) - 5

- 5 - 14 csc (x) + 3 csc^{2} (x) = 0

Solve by substitution

- 5 - 14 csc (x) + 3 csc^{2} (x) = 0

Let:

csc (x) = u

- 5 - 14 u + 3 u^{2} = 0

- 5 - 14 u + 3 u^{2} = 0 : u = 5, u = - \frac{1}{3}

- 5 - 14 u + 3 u^{2} = 0

Write in the standard form

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

3 u^{2} - 14 u - 5 = 0

Solve with the quadratic formula

3 u^{2} - 14 u - 5 = 0

Quadratic Equation Formula:

For

a = 3, b = - 14, c = - 5

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 3 ( - 5 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 3 ( - 5 )}{2 \cdot 3}

(- 14)^{2} - 4 \cdot 3 (- 5) = 16

(- 14)^{2} - 4 \cdot 3 (- 5)

Apply rule

- (- a) = a

= (- 14)^{2} + 4 \cdot 3 \cdot 5

Apply exponent rule:

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

n

is even

(- 14)^{2} = 1 4^{2}

= 1 4^{2} + 4 \cdot 3 \cdot 5

Multiply the numbers:

4 \cdot 3 \cdot 5 = 60

= 1 4^{2} + 60

1 4^{2} = 196

= 196 + 60

Add the numbers:

196 + 60 = 256

= 256

Factor the number:

256 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- ( - 14 ) \pm 16}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 14 ) + 16}{2 \cdot 3}, u_{2} = \frac{- ( - 14 ) - 16}{2 \cdot 3}

u = \frac{- ( - 14 ) + 16}{2 \cdot 3} : 5

\frac{- ( - 14 ) + 16}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{14 + 16}{2 \cdot 3}

Add the numbers:

14 + 16 = 30

= \frac{30}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{30}{6}

Divide the numbers:

\frac{30}{6} = 5

= 5

u = \frac{- ( - 14 ) - 16}{2 \cdot 3} : - \frac{1}{3}

\frac{- ( - 14 ) - 16}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{14 - 16}{2 \cdot 3}

Subtract the numbers:

14 - 16 = - 2

= \frac{- 2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 2}{6}

Apply the fraction rule:

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

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

The solutions to the quadratic equation are:

u = 5, u = - \frac{1}{3}

Substitute back

u = csc (x)

csc (x) = 5, csc (x) = - \frac{1}{3}

csc (x) = 5, csc (x) = - \frac{1}{3}

csc (x) = 5 : x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 5

Apply trig inverse properties

csc (x) = 5

General solutions for

csc (x) = 5

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

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

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

No Solution

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

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

No Solution

Combine all the solutions

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

Show solutions in decimal form

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

Popular Examples

250=100*5*cos(x)

250 = 100 \cdot 5 \cdot cos (x)

cos(θ)=-sqrt(2/5)

cos (θ) = - \frac{2}{5}

tan(x)= 1/(sin(2x)),0<= x<= pi

tan (x) = \frac{1}{sin ( 2 x )}, 0 \leq x \leq π

cos(x)=9054

cos (x) = 9054

1+cos(2x)=2sin^2(x)

1 + cos (2 x) = 2 sin^{2} (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(x)-14csc(x)-2=0 ?
The general solution for 3cot^2(x)-14csc(x)-2=0 is x=0.20135…+2pin,x=pi-0.20135…+2pin