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arctan(x/(12))-arctan(x)=0.001

Solution

arctan (\frac{x}{12}) - arctan (x) = 0.001

Solution

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Solution steps

arctan (\frac{x}{12}) - arctan (x) = 0.001

Rewrite using trig identities

arctan (\frac{x}{12}) - arctan (x)

Use the Sum to Product identity:

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

= arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x})

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

Apply trig inverse properties

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

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

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (0.001)

tan (0.001) = tan (\frac{1}{1000})

tan (0.001)

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

Solve

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

Simplify

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} : - \frac{11 x}{12 + x ^{2}}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}

\frac{x}{12} x = \frac{x ^{2}}{12}

\frac{x}{12} x

Multiply fractions:

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

= \frac{xx}{12}

xx = x^{2}

xx

Apply exponent rule:

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

xx = x^{1 + 1}

= x^{1 + 1}

Add the numbers:

1 + 1 = 2

= x^{2}

= \frac{x ^{2}}{12}

= \frac{\frac{x}{12} - x}{1 + \frac{x ^{2}}{12}}

Join

\frac{x}{12} - x : - \frac{11 x}{12}

\frac{x}{12} - x

Convert element to fraction:

x = \frac{x 12}{12}

= \frac{x}{12} - \frac{x \cdot 12}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{x - x \cdot 12}{12}

Add similar elements:

x - 12 x = - 11 x

= \frac{- 11 x}{12}

Apply the fraction rule:

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

= - \frac{11 x}{12}

= \frac{- \frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

Apply the fraction rule:

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

= - \frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

Apply the fraction rule:

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

\frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}} = \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

= - \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

Join

1 + \frac{x ^{2}}{12} : \frac{12 + x ^{2}}{12}

1 + \frac{x ^{2}}{12}

Convert element to fraction:

1 = \frac{1 \cdot 12}{12}

= \frac{1 \cdot 12}{12} + \frac{x ^{2}}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 12 + x ^{2}}{12}

Multiply the numbers:

1 \cdot 12 = 12

= \frac{12 + x ^{2}}{12}

= - \frac{11 x}{12 \cdot \frac{x ^{2} + 12}{12}}

Multiply

12 \cdot \frac{12 + x ^{2}}{12} : 12 + x^{2}

12 \cdot \frac{12 + x ^{2}}{12}

Multiply fractions:

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

= \frac{( 12 + x ^{2} ) \cdot 12}{12}

Cancel the common factor:

12

= 12 + x^{2}

= - \frac{11 x}{x ^{2} + 12}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

Multiply both sides by

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

Multiply both sides by

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} (12 + x^{2}) = tan (\frac{1}{1000}) (12 + x^{2})

Simplify

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

Solve

- 11 x = tan (\frac{1}{1000}) (12 + x^{2}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

Expand

tan (\frac{1}{1000}) (12 + x^{2}) : 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

tan (\frac{1}{1000}) (12 + x^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = tan (\frac{1}{1000}), b = 12, c = x^{2}

= tan (\frac{1}{1000}) \cdot 12 + tan (\frac{1}{1000}) x^{2}

= 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

- 11 x = 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

Switch sides

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

Move

11 x

to the left side

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

Add

11 x

to both sides

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = - 11 x + 11 x

Simplify

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

Write in the standard form

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

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

Solve with the quadratic formula

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

Quadratic Equation Formula:

For

a = tan (\frac{1}{1000}), b = 11, c = 12 tan (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 121 - 48 tan^{2} (\frac{1}{1000})

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 48 tan^{2} (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

Multiply the numbers:

4 \cdot 12 = 48

= 48 tan (\frac{1}{1000}) tan (\frac{1}{1000})

Apply exponent rule:

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

tan (\frac{1}{1000}) tan (\frac{1}{1000}) = tan^{1 + 1} (\frac{1}{1000})

= 48 tan^{1 + 1} (\frac{1}{1000})

Add the numbers:

1 + 1 = 2

= 48 tan^{2} (\frac{1}{1000})

= 1 1^{2} - 48 tan^{2} (\frac{1}{1000})

1 1^{2} = 121

= 121 - 48 tan^{2} (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Separate the solutions

x_{1} = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x_{2} = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

The solutions to the quadratic equation are:

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (\frac{x}{12}) - arctan (x) = 0.001

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

Check the solution

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} :

True

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Plug in

n = 1

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

For

arctan (\frac{x}{12}) - arctan (x) = 0.001

plug in

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 + 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

Refine

0.00099 \dots = 0.001

\Rightarrow True

Check the solution

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} :

True

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

Plug in

n = 1

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

For

arctan (\frac{x}{12}) - arctan (x) = 0.001

plug in

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 - 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

Refine

0.001 = 0.001

\Rightarrow True

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

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