What is the general solution for 5tan(x)=15cot(x) ?

The general solution for 5tan(x)=15cot(x) is x=1.04719…+pin,x=2.09439…+pin

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5tan(x)=15cot(x)

Solution

5 tan (x) = 15 cot (x)

Solution

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

5 tan (x) = 15 cot (x)

Subtract

15 cot (x)

from both sides

5 tan (x) - 15 cot (x) = 0

Rewrite using trig identities

- 15 cot (x) + 5 tan (x)

Use the basic trigonometric identity:

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

= - 15 cot (x) + 5 \cdot \frac{1}{cot ( x )}

5 \cdot \frac{1}{cot ( x )} = \frac{5}{cot ( x )}

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

Multiply fractions:

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

= \frac{1 \cdot 5}{cot ( x )}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{cot ( x )}

= - 15 cot (x) + \frac{5}{cot ( x )}

\frac{5}{cot ( x )} - 15 cot (x) = 0

Solve by substitution

\frac{5}{cot ( x )} - 15 cot (x) = 0

Let:

cot (x) = u

\frac{5}{u} - 15 u = 0

\frac{5}{u} - 15 u = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

\frac{5}{u} - 15 u = 0

Multiply both sides by

u

\frac{5}{u} - 15 u = 0

Multiply both sides by

u

\frac{5}{u} u - 15 uu = 0 \cdot u

Simplify

\frac{5}{u} u - 15 uu = 0 \cdot u

Simplify

\frac{5}{u} u : 5

\frac{5}{u} u

Multiply fractions:

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

= \frac{5 u}{u}

Cancel the common factor:

u

= 5

Simplify

- 15 uu : - 15 u^{2}

- 15 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 15 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 15 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

5 - 15 u^{2} = 0

5 - 15 u^{2} = 0

5 - 15 u^{2} = 0

Solve

5 - 15 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

5 - 15 u^{2} = 0

Move

5

to the right side

5 - 15 u^{2} = 0

Subtract

5

from both sides

5 - 15 u^{2} - 5 = 0 - 5

Simplify

- 15 u^{2} = - 5

- 15 u^{2} = - 5

Divide both sides by

- 15

- 15 u^{2} = - 5

Divide both sides by

- 15

\frac{- 15 u ^{2}}{- 15} = \frac{- 5}{- 15}

Simplify

\frac{- 15 u ^{2}}{- 15} = \frac{- 5}{- 15}

Simplify

\frac{- 15 u ^{2}}{- 15} : u^{2}

\frac{- 15 u ^{2}}{- 15}

Apply the fraction rule:

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

= \frac{15 u ^{2}}{15}

Divide the numbers:

\frac{15}{15} = 1

= u^{2}

Simplify

\frac{- 5}{- 15} : \frac{1}{3}

\frac{- 5}{- 15}

Apply the fraction rule:

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

= \frac{5}{15}

Cancel the common factor:

5

= \frac{1}{3}

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

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{5}{u} - 15 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = cot (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (\frac{1}{3}) + πn

x = arccot (\frac{1}{3}) + πn

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (- \frac{1}{3}) + πn

x = arccot (- \frac{1}{3}) + πn

Combine all the solutions

x = arccot (\frac{1}{3}) + πn, x = arccot (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5tan(x)=15cot(x) ?
The general solution for 5tan(x)=15cot(x) is x=1.04719…+pin,x=2.09439…+pin