What is the general solution for tan^2(θ)-3cot(θ)=0 ?

The general solution for tan^2(θ)-3cot(θ)=0 is θ=0.96453…+pin

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tan^2(θ)-3cot(θ)=0

Solution

tan^{2} (θ) - 3 cot (θ) = 0

Solution

θ = 0.96453 \dots + πn

Degrees

θ = 55.26405 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (θ) - 3 cot (θ) = 0

Rewrite using trig identities

tan^{2} (θ) - 3 cot (θ)

Use the basic trigonometric identity:

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

= (\frac{1}{cot ( θ )})^{2} - 3 cot (θ)

(\frac{1}{cot ( θ )})^{2} = \frac{1}{cot ^{2} ( θ )}

(\frac{1}{cot ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cot ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( θ )}

= \frac{1}{cot ^{2} ( θ )} - 3 cot (θ)

\frac{1}{cot ^{2} ( θ )} - 3 cot (θ) = 0

Solve by substitution

\frac{1}{cot ^{2} ( θ )} - 3 cot (θ) = 0

Let:

cot (θ) = u

\frac{1}{u ^{2}} - 3 u = 0

\frac{1}{u ^{2}} - 3 u = 0

Multiply both sides by

u^{2}

\frac{1}{u ^{2}} - 3 u = 0

Multiply both sides by

u^{2}

\frac{1}{u ^{2}} u^{2} - 3 u u^{2} = 0 \cdot u^{2}

Simplify

\frac{1}{u ^{2}} u^{2} - 3 u u^{2} = 0 \cdot u^{2}

Simplify

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1

Simplify

- 3 u u^{2} : - 3 u^{3}

- 3 u u^{2}

Apply exponent rule:

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

u u^{2} = u^{1 + 2}

= - 3 u^{1 + 2}

Add the numbers:

1 + 2 = 3

= - 3 u^{3}

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

1 - 3 u^{3} = 0

1 - 3 u^{3} = 0

1 - 3 u^{3} = 0

Solve

1 - 3 u^{3} = 0

Move

1

to the right side

1 - 3 u^{3} = 0

Subtract

1

from both sides

1 - 3 u^{3} - 1 = 0 - 1

Simplify

- 3 u^{3} = - 1

- 3 u^{3} = - 1

Divide both sides by

- 3

- 3 u^{3} = - 1

Divide both sides by

- 3

\frac{- 3 u ^{3}}{- 3} = \frac{- 1}{- 3}

Simplify

u^{3} = \frac{1}{3}

u^{3} = \frac{1}{3}

For

x^{3} = f (a)

the solutions are

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

Multiply:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

Remove parentheses:

(- a) = - a

= - 1 + 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

Apply exponent rule:

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

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

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

3^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

Rewrite

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

in standard complex form:

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3 i}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

Subtract the numbers:

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

= \frac{- 1 + 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

= \frac{- 1 + 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

Apply radical rule:

Apply the fraction rule:

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

Cancel

Apply radical rule:

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

Apply exponent rule:

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

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

= \frac{3 ^{\frac{1}{2} - \frac{1}{3}} i}{2}

Subtract the numbers:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

Apply radical rule:

Multiply by the conjugate

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

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

Apply exponent rule:

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

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

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

3^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

Multiply:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

Remove parentheses:

(- a) = - a

= - 1 - 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

Apply exponent rule:

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

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

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

3^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

Rewrite

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

in standard complex form:

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3}

Cancel

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3} : \frac{- 1 - 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 - 3 i}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

Subtract the numbers:

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

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

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

Apply radical rule:

Apply the fraction rule:

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

Cancel

Apply radical rule:

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

Apply exponent rule:

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

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

= \frac{3 ^{\frac{1}{2} - \frac{1}{3}} i}{2}

Subtract the numbers:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

Apply radical rule:

Multiply by the conjugate

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

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

Apply exponent rule:

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

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

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

3^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{1}{u ^{2}} - 3 u

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

Substitute back

u = cot (θ)

Apply trig inverse properties

General solutions for

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

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

θ = 0.96453 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(θ)-3cot(θ)=0 ?
The general solution for tan^2(θ)-3cot(θ)=0 is θ=0.96453…+pin