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

The general solution for tan(x)+1=-1/(sqrt(3))-1/(sqrt(3))cot(x) is x=(3pi)/4+pin,x=(5pi)/6+pin

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tan(x)+1=-1/(sqrt(3))-1/(sqrt(3))cot(x)

Solution

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

Solution

x = \frac{3 π}{4} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

tan (x) + 1 - \frac{- 1 - cot ( x )}{3} = 0

Simplify

tan (x) + 1 - \frac{- 1 - cot ( x )}{3} : \frac{3 tan ( x ) + 3 + 1 + cot ( x )}{3}

tan (x) + 1 - \frac{- 1 - cot ( x )}{3}

Convert element to fraction:

tan (x) = \frac{tan ( x ) 3}{3}, 1 = \frac{1 3}{3}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Multiply:

1 \cdot 3 = 3

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

Expand

tan (x) 3 + 3 - (- 1 - cot (x)) : tan (x) 3 + 3 + 1 + cot (x)

tan (x) 3 + 3 - (- 1 - cot (x))

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

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

- (- 1 - cot (x))

Distribute parentheses

= - (- 1) - (- cot (x))

Apply minus-plus rules

- (- a) = a

= 1 + cot (x)

= tan (x) 3 + 3 + 1 + cot (x)

= \frac{3 tan ( x ) + 3 + 1 + cot ( x )}{3}

\frac{3 tan ( x ) + 3 + 1 + cot ( x )}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

1 + cot (x) + 3 + 3 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

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

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Multiply:

1 \cdot 3 = 3

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

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

1 + cot (x) + \frac{3}{cot ( x )} + 3 = 0

Solve by substitution

1 + cot (x) + \frac{3}{cot ( x )} + 3 = 0

Let:

cot (x) = u

1 + u + \frac{3}{u} + 3 = 0

1 + u + \frac{3}{u} + 3 = 0 : u = - 1, u = - 3

1 + u + \frac{3}{u} + 3 = 0

Multiply both sides by

u

1 + u + \frac{3}{u} + 3 = 0

Multiply both sides by

u

1 \cdot u + uu + \frac{3}{u} u + 3 u = 0 \cdot u

Simplify

1 \cdot u + uu + \frac{3}{u} u + 3 u = 0 \cdot u

Simplify

1 \cdot u : u

1 \cdot u

Multiply:

1 \cdot u = u

= u

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{3}{u} u : 3

\frac{3}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 u}{u}

Cancel the common factor:

u

= 3

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u + u^{2} + 3 + 3 u = 0

u + u^{2} + 3 + 3 u = 0

u + u^{2} + 3 + 3 u = 0

Solve

u + u^{2} + 3 + 3 u = 0 : u = - 1, u = - 3

u + u^{2} + 3 + 3 u = 0

Write in the standard form

a x^{2} + b x + c = 0

u^{2} + (1 + 3) u + 3 = 0

Solve with the quadratic formula

u^{2} + (1 + 3) u + 3 = 0

Quadratic Equation Formula:

For

a = 1, b = 1 + 3, c = 3

u_{1, 2} = \frac{- ( 1 + 3 ) \pm ( 1 + 3 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- ( 1 + 3 ) \pm ( 1 + 3 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

(1 + 3)^{2} - 4 \cdot 1 \cdot 3 = 3 - 1

(1 + 3)^{2} - 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 = 4

= (1 + 3)^{2} - 43

Expand

(1 + 3)^{2} - 43 : 4 - 23

(1 + 3)^{2} - 43

(1 + 3)^{2} : 4 + 23

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 1, b = 3

= 1^{2} + 2 \cdot 1 \cdot 3 + (3)^{2}

Simplify

1^{2} + 2 \cdot 1 \cdot 3 + (3)^{2} : 4 + 23

1^{2} + 2 \cdot 1 \cdot 3 + (3)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 + 2 \cdot 1 \cdot 3 + (3)^{2}

2 \cdot 1 \cdot 3 = 23

2 \cdot 1 \cdot 3

Multiply the numbers:

2 \cdot 1 = 2

= 23

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 1 + 23 + 3

Add the numbers:

1 + 3 = 4

= 4 + 23

= 4 + 23

= 4 + 23 - 43

Add similar elements:

23 - 43 = - 23

= 4 - 23

= 4 - 23

= 3 - 23 + 1

= (3)^{2} - 23 + (1)^{2}

1 = 1

1

Apply rule

1 = 1

= 1

= (3)^{2} - 23 + 1^{2}

23 \cdot 1 = 23

23 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 23

= (3)^{2} - 23 \cdot 1 + 1^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

(3)^{2} - 23 \cdot 1 + 1^{2} = (3 - 1)^{2}

= (3 - 1)^{2}

Apply radical rule:

(3 - 1)^{2} = 3 - 1

= 3 - 1

u_{1, 2} = \frac{- ( 1 + 3 ) \pm ( 3 - 1 )}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( 1 + 3 ) + 3 - 1}{2 \cdot 1}, u_{2} = \frac{- ( 1 + 3 ) - ( 3 - 1 )}{2 \cdot 1}

u = \frac{- ( 1 + 3 ) + 3 - 1}{2 \cdot 1} : - 1

\frac{- ( 1 + 3 ) + 3 - 1}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- ( 1 + 3 ) + 3 - 1}{2}

Expand

- (1 + 3) + 3 - 1 : - 2

- (1 + 3) + 3 - 1

- (1 + 3) : - 1 - 3

- (1 + 3)

Distribute parentheses

= - (1) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 3

= - 1 - 3 + 3 - 1

Simplify

- 1 - 3 + 3 - 1 : - 2

- 1 - 3 + 3 - 1

Add similar elements:

- 3 + 3 = 0

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

= - 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- ( 1 + 3 ) - ( 3 - 1 )}{2 \cdot 1} : - 3

\frac{- ( 1 + 3 ) - ( 3 - 1 )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- ( 1 + 3 ) - ( 3 - 1 )}{2}

Expand

- (1 + 3) - (3 - 1) : - 23

- (1 + 3) - (3 - 1)

- (1 + 3) : - 1 - 3

- (1 + 3)

Distribute parentheses

= - (1) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 3

= - 1 - 3 - (3 - 1)

- (3 - 1) : - 3 + 1

- (3 - 1)

Distribute parentheses

= - (3) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 3 + 1

= - 1 - 3 - 3 + 1

Simplify

- 1 - 3 - 3 + 1 : - 23

- 1 - 3 - 3 + 1

Add similar elements:

- 3 - 3 = - 23

= - 1 - 23 + 1

- 1 + 1 = 0

= - 23

= - 23

= \frac{- 2 3}{2}

Apply the fraction rule:

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

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

The solutions to the quadratic equation are:

u = - 1, u = - 3

u = - 1, u = - 3

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 + u + \frac{3}{u} + 3

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1, u = - 3

Substitute back

u = cot (x)

cot (x) = - 1, cot (x) = - 3

cot (x) = - 1, cot (x) = - 3

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

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

cot (x) = - 3 : x = \frac{5 π}{6} + πn

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{5 π}{6} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+1=-1/(sqrt(3))-1/(sqrt(3))cot(x) ?
The general solution for tan(x)+1=-1/(sqrt(3))-1/(sqrt(3))cot(x) is x=(3pi)/4+pin,x=(5pi)/6+pin