What is the general solution for 2cot(x)cos(x)+2cos(x)=cot(x)+1 ?

The general solution for 2cot(x)cos(x)+2cos(x)=cot(x)+1 is x=(3pi)/4+pin,x= pi/3+2pin,x=(5pi)/3+2pin

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Popular Trigonometry >

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

Solution

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

Solution

x = \frac{3 π}{4} + πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cot (x) + 1

from both sides

2 cot (x) cos (x) + 2 cos (x) - cot (x) - 1 = 0

Factor

2 cot (x) cos (x) + 2 cos (x) - cot (x) - 1 : (cot (x) + 1) (2 cos (x) - 1)

2 cot (x) cos (x) + 2 cos (x) - cot (x) - 1

= (2 cot (x) cos (x) + 2 cos (x)) + (- cot (x) - 1)

Factor out

- 1

from

- cot (x) - 1 : - (cot (x) + 1)

- cot (x) - 1

Factor out common term

- 1

= - (cot (x) + 1)

Factor out

2 cos (x)

from

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

2 cot (x) cos (x) + 2 cos (x)

Factor out common term

2 cos (x)

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

= - (cot (x) + 1) + 2 cos (x) (cot (x) + 1)

Factor out common term

cot (x) + 1

= (cot (x) + 1) (2 cos (x) - 1)

(cot (x) + 1) (2 cos (x) - 1) = 0

Solving each part separately

cot (x) + 1 = 0 or 2 cos (x) - 1 = 0

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

cot (x) + 1 - 1 = 0 - 1

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 cos (x) - 1 = 0

Move

1

to the right side

2 cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 2cot(x)cos(x)+2cos(x)=cot(x)+1 ?
The general solution for 2cot(x)cos(x)+2cos(x)=cot(x)+1 is x=(3pi)/4+pin,x= pi/3+2pin,x=(5pi)/3+2pin