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

The general solution for sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ) is θ= pi/6+pin

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sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ)

Solution

3 sin (θ) cot (θ) - 2 sin (θ) = sin (θ)

Solution

θ = \frac{π}{6} + πn

Degrees

θ = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (θ) cot (θ) - 2 sin (θ) = sin (θ)

Subtract

sin (θ)

from both sides

3 sin (θ) cot (θ) - 3 sin (θ) = 0

Factor

3 sin (θ) cot (θ) - 3 sin (θ) : sin (θ) (3 cot (θ) - 3)

3 sin (θ) cot (θ) - 3 sin (θ)

Factor out common term

sin (θ)

= sin (θ) (3 cot (θ) - 3)

sin (θ) (3 cot (θ) - 3) = 0

Solving each part separately

sin (θ) = 0 or 3 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

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

3 cot (θ) - 3 = 0

Move

3

to the right side

3 cot (θ) - 3 = 0

Add

3

to both sides

3 cot (θ) - 3 + 3 = 0 + 3

Simplify

3 cot (θ) = 3

3 cot (θ) = 3

Divide both sides by

3

3 cot (θ) = 3

Divide both sides by

3

\frac{3 cot ( θ )}{3} = \frac{3}{3}

Simplify

\frac{3 cot ( θ )}{3} = \frac{3}{3}

Simplify

\frac{3 cot ( θ )}{3} : cot (θ)

\frac{3 cot ( θ )}{3}

Cancel the common factor:

3

= cot (θ)

Simplify

\frac{3}{3} : 3

\frac{3}{3}

Apply radical rule:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3^{\frac{1}{2}}

Apply radical rule:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= 3

cot (θ) = 3

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{π}{6} + πn

Since the equation is undefined for:

2 πn, π + 2 πn

θ = \frac{π}{6} + πn

Graph

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Frequently Asked Questions (FAQ)

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