What is the general solution for 2+3cot^2(θ)-3csc(θ)=5 ?

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

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2+3cot^2(θ)-3csc(θ)=5

Solution

2 + 3 cot^{2} (θ) - 3 csc (θ) = 5

Solution

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{3 π}{2} + 2 πn

Degrees

θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

2 + 3 cot^{2} (θ) - 3 csc (θ) = 5

Subtract

5

from both sides

3 cot^{2} (θ) - 3 csc (θ) - 3 = 0

Rewrite using trig identities

- 3 + 3 cot^{2} (θ) - 3 csc (θ)

Use the Pythagorean identity:

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

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

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

Simplify

- 3 + 3 (csc^{2} (θ) - 1) - 3 csc (θ) : 3 csc^{2} (θ) - 3 csc (θ) - 6

- 3 + 3 (csc^{2} (θ) - 1) - 3 csc (θ)

Expand

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

3 (csc^{2} (θ) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = csc^{2} (θ), c = 1

= 3 csc^{2} (θ) - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

= 3 csc^{2} (θ) - 3

= - 3 + 3 csc^{2} (θ) - 3 - 3 csc (θ)

Simplify

- 3 + 3 csc^{2} (θ) - 3 - 3 csc (θ) : 3 csc^{2} (θ) - 3 csc (θ) - 6

- 3 + 3 csc^{2} (θ) - 3 - 3 csc (θ)

Group like terms

= 3 csc^{2} (θ) - 3 csc (θ) - 3 - 3

Subtract the numbers:

- 3 - 3 = - 6

= 3 csc^{2} (θ) - 3 csc (θ) - 6

= 3 csc^{2} (θ) - 3 csc (θ) - 6

= 3 csc^{2} (θ) - 3 csc (θ) - 6

- 6 - 3 csc (θ) + 3 csc^{2} (θ) = 0

Solve by substitution

- 6 - 3 csc (θ) + 3 csc^{2} (θ) = 0

Let:

csc (θ) = u

- 6 - 3 u + 3 u^{2} = 0

- 6 - 3 u + 3 u^{2} = 0 : u = 2, u = - 1

- 6 - 3 u + 3 u^{2} = 0

Write in the standard form

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

3 u^{2} - 3 u - 6 = 0

Solve with the quadratic formula

3 u^{2} - 3 u - 6 = 0

Quadratic Equation Formula:

For

a = 3, b = - 3, c = - 6

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

(- 3)^{2} - 4 \cdot 3 (- 6) = 9

(- 3)^{2} - 4 \cdot 3 (- 6)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 3 \cdot 6

Apply exponent rule:

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

n

is even

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

= 3^{2} + 4 \cdot 3 \cdot 6

Multiply the numbers:

4 \cdot 3 \cdot 6 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

Add the numbers:

9 + 72 = 81

= 81

Factor the number:

81 = 9^{2}

= 9^{2}

Apply radical rule:

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 3 ) \pm 9}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 9}{2 \cdot 3}, u_{2} = \frac{- ( - 3 ) - 9}{2 \cdot 3}

u = \frac{- ( - 3 ) + 9}{2 \cdot 3} : 2

\frac{- ( - 3 ) + 9}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 + 9}{2 \cdot 3}

Add the numbers:

3 + 9 = 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{- ( - 3 ) - 9}{2 \cdot 3} : - 1

\frac{- ( - 3 ) - 9}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 - 9}{2 \cdot 3}

Subtract the numbers:

3 - 9 = - 6

= \frac{- 6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6}{6}

Apply the fraction rule:

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

= - \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 2, u = - 1

Substitute back

u = csc (θ)

csc (θ) = 2, csc (θ) = - 1

csc (θ) = 2, csc (θ) = - 1

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

csc (θ) = 2

General solutions for

csc (θ) = 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

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

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

csc (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

csc (θ) = - 1

General solutions for

csc (θ) = - 1

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

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

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