What is the general solution for (1+cot(x))/(1+tan(x))=5 ?

The general solution for (1+cot(x))/(1+tan(x))=5 is x=0.19739…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(1+cot(x))/(1+tan(x))=5

Solution

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

Solution

x = 0.19739 \dots + πn

Degrees

x = 11.30993 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

5

from both sides

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

Simplify

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

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

Convert element to fraction:

5 = \frac{5 ( 1 + tan ( x ) )}{1 + tan ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

1 + cot (x) - 5 (1 + tan (x)) : cot (x) - 5 tan (x) - 4

1 + cot (x) - 5 (1 + tan (x))

Expand

- 5 (1 + tan (x)) : - 5 - 5 tan (x)

- 5 (1 + tan (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - 5, b = 1, c = tan (x)

= - 5 \cdot 1 + (- 5) tan (x)

Apply minus-plus rules

+ (- a) = - a

= - 5 \cdot 1 - 5 tan (x)

Multiply the numbers:

5 \cdot 1 = 5

= - 5 - 5 tan (x)

= 1 + cot (x) - 5 - 5 tan (x)

Simplify

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

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

Group like terms

= cot (x) - 5 tan (x) + 1 - 5

Add/Subtract the numbers:

1 - 5 = - 4

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

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

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= - 4 + 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 )}

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

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

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

Multiply the numbers:

4 \cdot 1 \cdot 5 = 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 \cdot 1}

Separate the solutions

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

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

\frac{- ( - 4 ) + 6}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{4 + 6}{2 \cdot 1}

Add the numbers:

4 + 6 = 10

= \frac{10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5

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

\frac{- ( - 4 ) - 6}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{4 - 6}{2 \cdot 1}

Subtract the numbers:

4 - 6 = - 2

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 5, u = - 1

u = 5, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 5, u = - 1

Substitute back

u = cot (x)

cot (x) = 5, cot (x) = - 1

cot (x) = 5, cot (x) = - 1

cot (x) = 5 : x = arccot (5) + πn

cot (x) = 5

Apply trig inverse properties

cot (x) = 5

General solutions for

cot (x) = 5

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

x = arccot (5) + πn

x = arccot (5) + πn

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

Combine all the solutions

x = arccot (5) + πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

\frac{3 π}{4} + πn

x = arccot (5) + πn

Show solutions in decimal form

x = 0.19739 \dots + πn

Popular Examples

4=4cos(x)

4 = 4 cos (x)

cos^2(x)=2sin(x)-2

cos^{2} (x) = 2 sin (x) - 2

-3=39.6862600000000…cos(x)

- 3 = 39.6862600000000 \dots cos (x)

1+cos(θ)=-cos(θ)

1 + cos (θ) = - cos (θ)

2cos^2(x)-cos(-x)-1=0

2 cos^{2} (x) - cos (- x) - 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for (1+cot(x))/(1+tan(x))=5 ?
The general solution for (1+cot(x))/(1+tan(x))=5 is x=0.19739…+pin