What is the general solution for sin(θ)cot^2(θ)-3sin(θ)=0 ?

The general solution for sin(θ)cot^2(θ)-3sin(θ)=0 is θ=(5pi)/6+pin,θ= pi/6+pin

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sin(θ)cot^2(θ)-3sin(θ)=0

Solution

sin (θ) cot^{2} (θ) - 3 sin (θ) = 0

Solution

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

Degrees

θ = 15 0^{\circ} + 18 0^{\circ} n, θ = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) cot^{2} (θ) - 3 sin (θ) = 0

Factor

sin (θ) cot^{2} (θ) - 3 sin (θ) : sin (θ) (cot (θ) + 3) (cot (θ) - 3)

sin (θ) cot^{2} (θ) - 3 sin (θ)

Factor out common term

sin (θ)

= sin (θ) (cot^{2} (θ) - 3)

Factor

cot^{2} (θ) - 3 : (cot (θ) + 3) (cot (θ) - 3)

cot^{2} (θ) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= cot^{2} (θ) - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cot^{2} (θ) - (3)^{2} = (cot (θ) + 3) (cot (θ) - 3)

= (cot (θ) + 3) (cot (θ) - 3)

= sin (θ) (cot (θ) + 3) (cot (θ) - 3)

sin (θ) (cot (θ) + 3) (cot (θ) - 3) = 0

Solving each part separately

sin (θ) = 0 or cot (θ) + 3 = 0 or cot (θ) - 3 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

cot (θ) + 3 = 0 : θ = \frac{5 π}{6} + πn

cot (θ) + 3 = 0

Move

3

to the right side

cot (θ) + 3 = 0

Subtract

3

from both sides

cot (θ) + 3 - 3 = 0 - 3

Simplify

cot (θ) = - 3

cot (θ) = - 3

General solutions for

cot (θ) = - 3

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{5 π}{6} + πn

θ = \frac{5 π}{6} + πn

cot (θ) - 3 = 0 : θ = \frac{π}{6} + πn

cot (θ) - 3 = 0

Move

3

to the right side

cot (θ) - 3 = 0

Add

3

to both sides

cot (θ) - 3 + 3 = 0 + 3

Simplify

cot (θ) = 3

cot (θ) = 3

General solutions for

cot (θ) = 3

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{π}{6} + πn

θ = \frac{π}{6} + πn

Combine all the solutions

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

Since the equation is undefined for:

2 πn, π + 2 πn

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

Popular Examples

-6sin(x)-3=9

- 6 sin (x) - 3 = 9

5sin^2(x)+cos^2(x)=2

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

cos(θ)=-3/7

cos (θ) = - \frac{3}{7}

tan(x)cos(x)= 1/2

tan (x) cos (x) = \frac{1}{2}

sec(x)sin(x)-3sin(x)=0

sec (x) sin (x) - 3 sin (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)cot^2(θ)-3sin(θ)=0 ?
The general solution for sin(θ)cot^2(θ)-3sin(θ)=0 is θ=(5pi)/6+pin,θ= pi/6+pin