What is the general solution for tan(x)cot(x)-sqrt(3)tan(x)=0 ?

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

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tan(x)cot(x)-sqrt(3)tan(x)=0

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

x = \frac{π}{6} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

tan (x) cot (x) - 3 tan (x) : tan (x) (cot (x) - 3)

tan (x) cot (x) - 3 tan (x)

Factor out common term

tan (x)

= tan (x) (cot (x) - 3)

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

Solving each part separately

tan (x) = 0 or cot (x) - 3 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

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

cot (x) - 3 = 0

Move

3

to the right side

cot (x) - 3 = 0

Add

3

to both sides

cot (x) - 3 + 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{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = πn, x = \frac{π}{6} + πn

Since the equation is undefined for:

πn

x = \frac{π}{6} + πn

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