What is the general solution for 7cot(x)-sqrt(3)=4cot(x) ?

The general solution for 7cot(x)-sqrt(3)=4cot(x) is x= pi/3+pin

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7cot(x)-sqrt(3)=4cot(x)

7 cot (x) - 3 = 4 cot (x)

x = \frac{π}{3} + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

7 cot (x) - 3 = 4 cot (x)

Solve by substitution

7 cot (x) - 3 = 4 cot (x)

Let:

cot (x) = u

7 u - 3 = 4 u

7 u - 3 = 4 u : u = \frac{3}{3}

7 u - 3 = 4 u

Move

3

to the right side

7 u - 3 = 4 u

Add

3

to both sides

7 u - 3 + 3 = 4 u + 3

Simplify

7 u = 4 u + 3

7 u = 4 u + 3

Move

4 u

to the left side

7 u = 4 u + 3

Subtract

4 u

from both sides

7 u - 4 u = 4 u + 3 - 4 u

Simplify

3 u = 3

3 u = 3

Divide both sides by

3

3 u = 3

Divide both sides by

3

\frac{3 u}{3} = \frac{3}{3}

Simplify

u = \frac{3}{3}

u = \frac{3}{3}

Substitute back

u = cot (x)

cot (x) = \frac{3}{3}

cot (x) = \frac{3}{3}

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

cot (x) = \frac{3}{3}

General solutions for

cot (x) = \frac{3}{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}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

x = \frac{π}{3} + πn

cos (θ) + 2 cos^{2} (θ) = 1

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

sin^{2} (θ) - cos^{2} (θ) = 1 + cos (θ), 0 \leq θ < 2 π

csc (x) = \frac{1}{2}

sin^{2} (θ) - sin (θ) = 0

What is the general solution for 7cot(x)-sqrt(3)=4cot(x) ?
The general solution for 7cot(x)-sqrt(3)=4cot(x) is x= pi/3+pin