What is the general solution for sqrt(3)csc(x)+csc(x)cot(x)=0 ?

The general solution for sqrt(3)csc(x)+csc(x)cot(x)=0 is x=(5pi)/6+pin

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sqrt(3)csc(x)+csc(x)cot(x)=0

3 csc (x) + csc (x) cot (x) = 0

x = \frac{5 π}{6} + πn

Degrees

x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

3 csc (x) + csc (x) cot (x) = 0

Factor

3 csc (x) + csc (x) cot (x) : csc (x) (3 + cot (x))

3 csc (x) + csc (x) cot (x)

Factor out common term

csc (x)

= csc (x) (3 + cot (x))

csc (x) (3 + cot (x)) = 0

Solving each part separately

csc (x) = 0 or 3 + cot (x) = 0

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

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

3 + cot (x) = 0

Move

3

to the right side

3 + cot (x) = 0

Subtract

3

from both sides

3 + cot (x) - 3 = 0 - 3

Simplify

cot (x) = - 3

cot (x) = - 3

General solutions for

cot (x) = - 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

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

Combine all the solutions

x = \frac{5 π}{6} + πn

- 3 cos^{2} (θ) - 10 cos (θ) = - 3 cos (θ) - 6

- 4 sin (x) = - cos^{2} (x) + 4

sin^{2} (x) - 4 sin (x) + 3 = 0

6 sin (x) + csc (x) = 0

cot^{2} (x) - 3 csc (x) = - 3

What is the general solution for sqrt(3)csc(x)+csc(x)cot(x)=0 ?
The general solution for sqrt(3)csc(x)+csc(x)cot(x)=0 is x=(5pi)/6+pin