What is the general solution for 3cot(x)+1=2+4cot(x),0<= x<= 360 ?

Q: What is the general solution for 3cot(x)+1=2+4cot(x),0<= x<= 360 ?

The general solution for 3cot(x)+1=2+4cot(x),0<= x<= 360 is x=135,x=315

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3cot(x)+1=2+4cot(x),0<= x<= 360

Solution

3 cot (x) + 1 = 2 + 4 cot (x), 0^{\circ} \leq x \leq 36 0^{\circ}

Solution

x = 13 5^{\circ}, x = 31 5^{\circ}

Radians

x = \frac{3 π}{4}, x = \frac{7 π}{4}

Solution steps

3 cot (x) + 1 = 2 + 4 cot (x), 0^{\circ} \leq x \leq 36 0^{\circ}

Solve by substitution

3 cot (x) + 1 = 2 + 4 cot (x)

Let:

cot (x) = u

3 u + 1 = 2 + 4 u

3 u + 1 = 2 + 4 u : u = - 1

3 u + 1 = 2 + 4 u

Move

1

to the right side

3 u + 1 = 2 + 4 u

Subtract

1

from both sides

3 u + 1 - 1 = 2 + 4 u - 1

Simplify

3 u = 4 u + 1

3 u = 4 u + 1

Move

4 u

to the left side

3 u = 4 u + 1

Subtract

4 u

from both sides

3 u - 4 u = 4 u + 1 - 4 u

Simplify

- u = 1

- u = 1

Divide both sides by

- 1

- u = 1

Divide both sides by

- 1

\frac{- u}{- 1} = \frac{1}{- 1}

Simplify

u = - 1

u = - 1

Substitute back

u = cot (x)

cot (x) = - 1

cot (x) = - 1

cot (x) = - 1, 0 \leq x \leq 36 0^{\circ} : x = 13 5^{\circ}, x = 31 5^{\circ}

cot (x) = - 1, 0 \leq x \leq 36 0^{\circ}

General solutions for

cot (x) = - 1

cot (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

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

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

Solutions for the range

0 \leq x \leq 36 0^{\circ}

x = 13 5^{\circ}, x = 31 5^{\circ}

Combine all the solutions

x = 13 5^{\circ}, x = 31 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for 3cot(x)+1=2+4cot(x),0<= x<= 360 ?
The general solution for 3cot(x)+1=2+4cot(x),0<= x<= 360 is x=135,x=315