What is the general solution for 3cot^2(x)-4csc(x)=1 ?

The general solution for 3cot^2(x)-4csc(x)=1 is x= pi/6+2pin,x=(5pi)/6+2pin

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3cot^2(x)-4csc(x)=1

Solution

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

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

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

Simplify

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

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

Group like terms

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

Subtract the numbers:

- 1 - 3 = - 4

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

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

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

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

Solve by substitution

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

Let:

csc (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 3 \cdot 4 = 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 3}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 3}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 3}

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

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

Apply rule

- (- a) = a

= \frac{4 + 8}{2 \cdot 3}

Add the numbers:

4 + 8 = 12

= \frac{12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

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

\frac{- ( - 4 ) - 8}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{4 - 8}{2 \cdot 3}

Subtract the numbers:

4 - 8 = - 4

= \frac{- 4}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{6}

Apply the fraction rule:

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

= - \frac{4}{6}

Cancel the common factor:

2

= - \frac{2}{3}

The solutions to the quadratic equation are:

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

Substitute back

u = csc (x)

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

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

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

General solutions for

csc (x) = 2

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

No Solution

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

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

No Solution

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(x)-4csc(x)=1 ?
The general solution for 3cot^2(x)-4csc(x)=1 is x= pi/6+2pin,x=(5pi)/6+2pin