What is the general solution for (cos(x)-1)(sqrt(3)cot(x)+1)=0 ?

The general solution for (cos(x)-1)(sqrt(3)cot(x)+1)=0 is x=(2pi)/3+pin

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(cos(x)-1)(sqrt(3)cot(x)+1)=0

Solution

(cos (x) - 1) (3 cot (x) + 1) = 0

Solution

x = \frac{2 π}{3} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

(cos (x) - 1) (3 cot (x) + 1) = 0

Solving each part separately

cos (x) - 1 = 0 or 3 cot (x) + 1 = 0

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

3 cot (x) + 1 = 0 : x = \frac{2 π}{3} + πn

3 cot (x) + 1 = 0

Move

1

to the right side

3 cot (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

3 cot (x) = - 1

3 cot (x) = - 1

Divide both sides by

3

3 cot (x) = - 1

Divide both sides by

3

\frac{3 cot ( x )}{3} = \frac{- 1}{3}

Simplify

\frac{3 cot ( x )}{3} = \frac{- 1}{3}

Simplify

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

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

Cancel the common factor:

3

= cot (x)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

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

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{2 π}{3} + πn

x = \frac{2 π}{3} + πn

Combine all the solutions

x = 2 πn, x = \frac{2 π}{3} + πn

Since the equation is undefined for:

2 πn

x = \frac{2 π}{3} + πn

Popular Examples

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4 cot (x) = 3 csc (x)

solvefor x,sin(2x)-cos(x)=0 solve for

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

sin(θ)= 7/15

sin (θ) = \frac{7}{15}

tan^2(y)+3=2sec^2(y)

tan^{2} (y) + 3 = 2 sec^{2} (y)

tan(b+15)*cot(2b-5)=1

tan (b + 15) \cdot cot (2 b - 5) = 1

Frequently Asked Questions (FAQ)

What is the general solution for (cos(x)-1)(sqrt(3)cot(x)+1)=0 ?
The general solution for (cos(x)-1)(sqrt(3)cot(x)+1)=0 is x=(2pi)/3+pin