What is the general solution for arctan(1/4 x)-pi-arctan(x/(100))=-pi ?

The general solution for arctan(1/4 x)-pi-arctan(x/(100))=-pi is x=0

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arctan(1/4 x)-pi-arctan(x/(100))=-pi

Solution

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

Solution

x = 0

Solution steps

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

Rewrite using trig identities

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100})

Use the Sum to Product identity:

arctan (s) - arctan (t) = arctan (\frac{s - t}{1 + s t})

= - π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}})

- π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = - π

Add

π

to both sides

- π + arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) + π = - π + π

Simplify

arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = 0

Apply trig inverse properties

arctan (\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}) = 0

arctan (x) = a \Rightarrow x = tan (a)

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = tan (0)

tan (0) = 0

tan (0)

Use the following trivial identity:

tan (0) = 0

tan (0)

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

Solve

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0 : x = 0

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} = 0

Simplify

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}} : \frac{96 x}{400 + x ^{2}}

\frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{1}{4} x \frac{x}{100}}

\frac{1}{4} x \frac{x}{100} = \frac{x ^{2}}{400}

\frac{1}{4} x \frac{x}{100}

Multiply fractions:

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

= \frac{1 \cdot xx}{4 \cdot 100}

1 \cdot xx = x^{2}

1 \cdot xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 1 \cdot x^{1 + 1}

Refine

= x^{2}

= \frac{x ^{2}}{4 \cdot 100}

Multiply the numbers:

4 \cdot 100 = 400

= \frac{x ^{2}}{400}

= \frac{\frac{1}{4} x - \frac{x}{100}}{1 + \frac{x ^{2}}{400}}

Join

\frac{1}{4} x - \frac{x}{100} : \frac{6 x}{25}

\frac{1}{4} x - \frac{x}{100}

Multiply

\frac{1}{4} x : \frac{x}{4}

\frac{1}{4} x

Multiply fractions:

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

= \frac{1 \cdot x}{4}

Multiply:

1 \cdot x = x

= \frac{x}{4}

= \frac{x}{4} - \frac{x}{100}

Least Common Multiplier of

4, 100 : 100

4, 100

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

100 : 2 \cdot 2 \cdot 5 \cdot 5

100

100

divides by

2100 = 50 \cdot 2

= 2 \cdot 50

50

divides by

250 = 25 \cdot 2

= 2 \cdot 2 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5 \cdot 5

Multiply each factor the greatest number of times it occurs in either

4

100

= 2 \cdot 2 \cdot 5 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 5 \cdot 5 = 100

= 100

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

100

For

\frac{x}{4} :

multiply the denominator and numerator by

25

\frac{x}{4} = \frac{x \cdot 25}{4 \cdot 25} = \frac{x \cdot 25}{100}

= \frac{x \cdot 25}{100} - \frac{x}{100}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 25 - x}{100}

Add similar elements:

25 x - x = 24 x

= \frac{24 x}{100}

Cancel the common factor:

4

= \frac{6 x}{25}

= \frac{\frac{6 x}{25}}{1 + \frac{x ^{2}}{400}}

Apply the fraction rule:

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

= \frac{6 x}{25 ( 1 + \frac{x ^{2}}{400} )}

Join

1 + \frac{x ^{2}}{400} : \frac{400 + x ^{2}}{400}

1 + \frac{x ^{2}}{400}

Convert element to fraction:

1 = \frac{1 \cdot 400}{400}

= \frac{1 \cdot 400}{400} + \frac{x ^{2}}{400}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 400 + x ^{2}}{400}

Multiply the numbers:

1 \cdot 400 = 400

= \frac{400 + x ^{2}}{400}

= \frac{6 x}{25 \cdot \frac{x ^{2} + 400}{400}}

Multiply

25 \cdot \frac{400 + x ^{2}}{400} : \frac{400 + x ^{2}}{16}

25 \cdot \frac{400 + x ^{2}}{400}

Multiply fractions:

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

= \frac{( 400 + x ^{2} ) \cdot 25}{400}

Cancel the common factor:

25

= \frac{400 + x ^{2}}{16}

= \frac{6 x}{\frac{400 + x ^{2}}{16}}

Apply the fraction rule:

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

= \frac{6 x \cdot 16}{400 + x ^{2}}

Multiply the numbers:

6 \cdot 16 = 96

= \frac{96 x}{400 + x ^{2}}

\frac{96 x}{400 + x ^{2}} = 0

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

96 x = 0

Divide both sides by

96

96 x = 0

Divide both sides by

96

\frac{96 x}{96} = \frac{0}{96}

Simplify

x = 0

x = 0

x = 0

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

Remove the ones that don't agree with the equation.

Check the solution

0 :

True

0

Plug in

n = 1

0

For

arctan (\frac{1}{4} x) - π - arctan (\frac{x}{100}) = - π

plug in

x = 0

arctan (\frac{1}{4} \cdot 0) - π - arctan (\frac{0}{100}) = - π

Refine

- 3.14159 \dots = - 3.14159 \dots

\Rightarrow True

x = 0

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(1/4 x)-pi-arctan(x/(100))=-pi ?
The general solution for arctan(1/4 x)-pi-arctan(x/(100))=-pi is x=0