What is the general solution for 2cot^2(x)-9csc(x)=3 ?

The general solution for 2cot^2(x)-9csc(x)=3 is x=0.20135…+2pin,x=pi-0.20135…+2pin

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Popular Trigonometry >

2cot^2(x)-9csc(x)=3

Solution

2 cot^{2} (x) - 9 csc (x) = 3

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

2 cot^{2} (x) - 9 csc (x) = 3

Subtract

3

from both sides

2 cot^{2} (x) - 9 csc (x) - 3 = 0

Rewrite using trig identities

- 3 + 2 cot^{2} (x) - 9 csc (x)

Use the Pythagorean identity:

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

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

= - 3 + 2 (csc^{2} (x) - 1) - 9 csc (x)

Simplify

- 3 + 2 (csc^{2} (x) - 1) - 9 csc (x) : 2 csc^{2} (x) - 9 csc (x) - 5

- 3 + 2 (csc^{2} (x) - 1) - 9 csc (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

= - 3 + 2 csc^{2} (x) - 2 - 9 csc (x)

Simplify

- 3 + 2 csc^{2} (x) - 2 - 9 csc (x) : 2 csc^{2} (x) - 9 csc (x) - 5

- 3 + 2 csc^{2} (x) - 2 - 9 csc (x)

Group like terms

= 2 csc^{2} (x) - 9 csc (x) - 3 - 2

Subtract the numbers:

- 3 - 2 = - 5

= 2 csc^{2} (x) - 9 csc (x) - 5

= 2 csc^{2} (x) - 9 csc (x) - 5

= 2 csc^{2} (x) - 9 csc (x) - 5

- 5 + 2 csc^{2} (x) - 9 csc (x) = 0

Solve by substitution

- 5 + 2 csc^{2} (x) - 9 csc (x) = 0

Let:

csc (x) = u

- 5 + 2 u^{2} - 9 u = 0

- 5 + 2 u^{2} - 9 u = 0 : u = 5, u = - \frac{1}{2}

- 5 + 2 u^{2} - 9 u = 0

Write in the standard form

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

2 u^{2} - 9 u - 5 = 0

Solve with the quadratic formula

2 u^{2} - 9 u - 5 = 0

Quadratic Equation Formula:

For

a = 2, b = - 9, c = - 5

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

(- 9)^{2} - 4 \cdot 2 (- 5) = 11

(- 9)^{2} - 4 \cdot 2 (- 5)

Apply rule

- (- a) = a

= (- 9)^{2} + 4 \cdot 2 \cdot 5

Apply exponent rule:

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

n

is even

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 2 \cdot 5

Multiply the numbers:

4 \cdot 2 \cdot 5 = 40

= 9^{2} + 40

9^{2} = 81

= 81 + 40

Add the numbers:

81 + 40 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 9 ) \pm 11}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 9 ) + 11}{2 \cdot 2}, u_{2} = \frac{- ( - 9 ) - 11}{2 \cdot 2}

u = \frac{- ( - 9 ) + 11}{2 \cdot 2} : 5

\frac{- ( - 9 ) + 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{9 + 11}{2 \cdot 2}

Add the numbers:

9 + 11 = 20

= \frac{20}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{20}{4}

Divide the numbers:

\frac{20}{4} = 5

= 5

u = \frac{- ( - 9 ) - 11}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 9 ) - 11}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{9 - 11}{2 \cdot 2}

Subtract the numbers:

9 - 11 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = 5, u = - \frac{1}{2}

Substitute back

u = csc (x)

csc (x) = 5, csc (x) = - \frac{1}{2}

csc (x) = 5, csc (x) = - \frac{1}{2}

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}{2} :

No Solution

csc (x) = - \frac{1}{2}

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

Graph

Popular Examples

sin(2x)+sqrt(2)cos(x)=0

Frequently Asked Questions (FAQ)

What is the general solution for 2cot^2(x)-9csc(x)=3 ?
The general solution for 2cot^2(x)-9csc(x)=3 is x=0.20135…+2pin,x=pi-0.20135…+2pin