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

The general solution for 3tan(x)-3cot(x)-1=0 is x=2.43876…+pin,x=0.86797…+pin

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

Solution

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

Solution

x = 2.43876 \dots + πn, x = 0.86797 \dots + πn

Degrees

x = 139.73116 \dots^{\circ} + 18 0^{\circ} n, x = 49.73116 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

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

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

Solve by substitution

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

Let:

cot (x) = u

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

- 1 + \frac{3}{u} - 3 u = 0 : u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

- 1 \cdot u : - u

- 1 \cdot u

Multiply:

1 \cdot u = u

= - u

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

- 3 uu : - 3 u^{2}

- 3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

- u + 3 - 3 u^{2} = 0 : u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 3, b = - 1, c = 3

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

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

(- 1)^{2} - 4 (- 3) \cdot 3 = 37

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

Apply rule

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 3 \cdot 3 = 36

4 \cdot 3 \cdot 3

Multiply the numbers:

4 \cdot 3 \cdot 3 = 36

= 36

= 1 + 36

Add the numbers:

1 + 36 = 37

= 37

u_{1, 2} = \frac{- ( - 1 ) \pm 37}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 37}{2 ( - 3 )}, u_{2} = \frac{- ( - 1 ) - 37}{2 ( - 3 )}

u = \frac{- ( - 1 ) + 37}{2 ( - 3 )} : - \frac{1 + 37}{6}

\frac{- ( - 1 ) + 37}{2 ( - 3 )}

Remove parentheses:

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

= \frac{1 + 37}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{1 + 37}{- 6}

Apply the fraction rule:

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

= - \frac{1 + 37}{6}

u = \frac{- ( - 1 ) - 37}{2 ( - 3 )} : \frac{37 - 1}{6}

\frac{- ( - 1 ) - 37}{2 ( - 3 )}

Remove parentheses:

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

= \frac{1 - 37}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{1 - 37}{- 6}

Apply the fraction rule:

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

1 - 37 = - (37 - 1)

= \frac{37 - 1}{6}

The solutions to the quadratic equation are:

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

Substitute back

u = cot (x)

cot (x) = - \frac{1 + 37}{6}, cot (x) = \frac{37 - 1}{6}

cot (x) = - \frac{1 + 37}{6}, cot (x) = \frac{37 - 1}{6}

cot (x) = - \frac{1 + 37}{6} : x = arccot (- \frac{1 + 37}{6}) + πn

cot (x) = - \frac{1 + 37}{6}

Apply trig inverse properties

cot (x) = - \frac{1 + 37}{6}

General solutions for

cot (x) = - \frac{1 + 37}{6}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{1 + 37}{6}) + πn

x = arccot (- \frac{1 + 37}{6}) + πn

cot (x) = \frac{37 - 1}{6} : x = arccot (\frac{37 - 1}{6}) + πn

cot (x) = \frac{37 - 1}{6}

Apply trig inverse properties

cot (x) = \frac{37 - 1}{6}

General solutions for

cot (x) = \frac{37 - 1}{6}

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{37 - 1}{6}) + πn

x = arccot (\frac{37 - 1}{6}) + πn

Combine all the solutions

x = arccot (- \frac{1 + 37}{6}) + πn, x = arccot (\frac{37 - 1}{6}) + πn

Show solutions in decimal form

x = 2.43876 \dots + πn, x = 0.86797 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)-3cot(x)-1=0 ?
The general solution for 3tan(x)-3cot(x)-1=0 is x=2.43876…+pin,x=0.86797…+pin