What is the general solution for sin(θ)+sin(θ)cot(θ)=0 ?

The general solution for sin(θ)+sin(θ)cot(θ)=0 is θ=(3pi)/4+pin

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sin(θ)+sin(θ)cot(θ)=0

sin (θ) + sin (θ) cot (θ) = 0

θ = \frac{3 π}{4} + πn

Degrees

θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ) + sin (θ) cot (θ) = 0

Factor

sin (θ) + sin (θ) cot (θ) : sin (θ) (cot (θ) + 1)

sin (θ) + sin (θ) cot (θ)

Factor out common term

sin (θ)

= sin (θ) (1 + cot (θ))

sin (θ) (cot (θ) + 1) = 0

Solving each part separately

sin (θ) = 0 or cot (θ) + 1 = 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) + 1 = 0 : θ = \frac{3 π}{4} + πn

cot (θ) + 1 = 0

Move

1

to the right side

cot (θ) + 1 = 0

Subtract

1

from both sides

cot (θ) + 1 - 1 = 0 - 1

Simplify

cot (θ) = - 1

cot (θ) = - 1

General solutions for

cot (θ) = - 1

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{3 π}{4} + πn

θ = \frac{3 π}{4} + πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{3 π}{4} + πn

Since the equation is undefined for:

2 πn, π + 2 πn

θ = \frac{3 π}{4} + πn

sin (2 x) = - sin (- x)

cos (x) - 4 = sin (x) - 4

(2 cos (x) + 1) (2 sin (x) + 3) = 0

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

9 cos (x) + 9 sin (x) tan (x) = 18

What is the general solution for sin(θ)+sin(θ)cot(θ)=0 ?
The general solution for sin(θ)+sin(θ)cot(θ)=0 is θ=(3pi)/4+pin