What is the general solution for 4+tan(x)=5cot(x) ?

The general solution for 4+tan(x)=5cot(x) is x=1.76819…+pin,x= pi/4+pin

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4+tan(x)=5cot(x)

Solution

4 + tan (x) = 5 cot (x)

Solution

x = 1.76819 \dots + πn, x = \frac{π}{4} + πn

Degrees

x = 101.30993 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

4 + tan (x) = 5 cot (x)

Subtract

5 cot (x)

from both sides

4 + tan (x) - 5 cot (x) = 0

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= 4 + \frac{1}{cot ( x )} - 5 cot (x)

4 + \frac{1}{cot ( x )} - 5 cot (x) = 0

Solve by substitution

4 + \frac{1}{cot ( x )} - 5 cot (x) = 0

Let:

cot (x) = u

4 + \frac{1}{u} - 5 u = 0

4 + \frac{1}{u} - 5 u = 0 : u = - \frac{1}{5}, u = 1

4 + \frac{1}{u} - 5 u = 0

Multiply both sides by

u

4 + \frac{1}{u} - 5 u = 0

Multiply both sides by

u

4 u + \frac{1}{u} u - 5 uu = 0 \cdot u

Simplify

4 u + \frac{1}{u} u - 5 uu = 0 \cdot u

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= 1

Simplify

- 5 uu : - 5 u^{2}

- 5 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 5 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

4 u + 1 - 5 u^{2} = 0

4 u + 1 - 5 u^{2} = 0

4 u + 1 - 5 u^{2} = 0

Solve

4 u + 1 - 5 u^{2} = 0 : u = - \frac{1}{5}, u = 1

4 u + 1 - 5 u^{2} = 0

Write in the standard form

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

- 5 u^{2} + 4 u + 1 = 0

Solve with the quadratic formula

- 5 u^{2} + 4 u + 1 = 0

Quadratic Equation Formula:

For

a = - 5, b = 4, c = 1

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 5 ) \cdot 1}{2 ( - 5 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 5 ) \cdot 1}{2 ( - 5 )}

4^{2} - 4 (- 5) \cdot 1 = 6

4^{2} - 4 (- 5) \cdot 1

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 5 \cdot 1

Multiply the numbers:

4 \cdot 5 \cdot 1 = 20

= 4^{2} + 20

4^{2} = 16

= 16 + 20

Add the numbers:

16 + 20 = 36

= 36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

u_{1, 2} = \frac{- 4 \pm 6}{2 ( - 5 )}

Separate the solutions

u_{1} = \frac{- 4 + 6}{2 ( - 5 )}, u_{2} = \frac{- 4 - 6}{2 ( - 5 )}

u = \frac{- 4 + 6}{2 ( - 5 )} : - \frac{1}{5}

\frac{- 4 + 6}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 6}{- 2 \cdot 5}

Add/Subtract the numbers:

- 4 + 6 = 2

= \frac{2}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{2}{- 10}

Apply the fraction rule:

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

= - \frac{2}{10}

Cancel the common factor:

2

= - \frac{1}{5}

u = \frac{- 4 - 6}{2 ( - 5 )} : 1

\frac{- 4 - 6}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 6}{- 2 \cdot 5}

Subtract the numbers:

- 4 - 6 = - 10

= \frac{- 10}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{- 10}

Apply the fraction rule:

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

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{1}{5}, u = 1

u = - \frac{1}{5}, u = 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

4 + \frac{1}{u} - 5 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{1}{5}, u = 1

Substitute back

u = cot (x)

cot (x) = - \frac{1}{5}, cot (x) = 1

cot (x) = - \frac{1}{5}, cot (x) = 1

cot (x) = - \frac{1}{5} : x = arccot (- \frac{1}{5}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (- \frac{1}{5}) + πn

x = arccot (- \frac{1}{5}) + πn

cot (x) = 1 : x = \frac{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{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = arccot (- \frac{1}{5}) + πn, x = \frac{π}{4} + πn

Show solutions in decimal form

x = 1.76819 \dots + πn, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4+tan(x)=5cot(x) ?
The general solution for 4+tan(x)=5cot(x) is x=1.76819…+pin,x= pi/4+pin