What is the general solution for 2cot(x)sin(x)+cot(x)=0 ?

The general solution for 2cot(x)sin(x)+cot(x)=0 is x= pi/2+pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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2cot(x)sin(x)+cot(x)=0

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

x = \frac{π}{2} + πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

2 cot (x) sin (x) + cot (x) : cot (x) (2 sin (x) + 1)

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

Factor out common term

cot (x)

= cot (x) (2 sin (x) + 1)

cot (x) (2 sin (x) + 1) = 0

Solving each part separately

cot (x) = 0 or 2 sin (x) + 1 = 0

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

General solutions for

cot (x) = 0

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{π}{2} + πn

x = \frac{π}{2} + πn

2 sin (x) + 1 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (x) + 1 = 0

Move

1

to the right side

2 sin (x) + 1 = 0

Subtract

1

from both sides

2 sin (x) + 1 - 1 = 0 - 1

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

Simplify

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

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 = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (x) - (2 + 3) = - 3 csc (x)

7 cos^{2} (θ) + 6 sin (θ) - 10 = - 4

3 (2 sin (x) - cos (x)) = 2 (sin (x) - 3 cos (x))

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

tan (θ) = \frac{7}{6}

What is the general solution for 2cot(x)sin(x)+cot(x)=0 ?
The general solution for 2cot(x)sin(x)+cot(x)=0 is x= pi/2+pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin