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

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

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sin(x)+sin(x)cot(x)=0

sin (x) + sin (x) cot (x) = 0

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) + sin (x) cot (x) = 0

Factor

sin (x) + sin (x) cot (x) : sin (x) (cot (x) + 1)

sin (x) + sin (x) cot (x)

Factor out common term

sin (x)

= sin (x) (1 + cot (x))

sin (x) (cot (x) + 1) = 0

Solving each part separately

sin (x) = 0 or cot (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

cot (x) + 1 - 1 = 0 - 1

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 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

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{3 π}{4} + πn

arctan (θ) = 1

cos (2 θ) = - \frac{1}{2}, 0 \leq θ \leq 2 π

sin (2 x) = \frac{2}{5}

sin (2 x) + cos (x) = 0, 0 \leq x \leq 2 π

cos (θ) = - 0.5

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