What is the general solution for arctan(0.1x)+arctan(0.01x)=39 ?

The general solution for arctan(0.1x)+arctan(0.01x)=39 is x=(sqrt(0.01472…)-0.11)/(0.00161…)

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arctan(0.1x)+arctan(0.01x)=39

Solution

arctan (0.1 x) + arctan (0.01 x) = 3 9^{\circ}

Solution

x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

Solution steps

arctan (0.1 x) + arctan (0.01 x) = 3 9^{\circ}

Rewrite using trig identities

arctan (0.1 x) + arctan (0.01 x)

Use the Sum to Product identity:

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

= arctan (\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x})

arctan (\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x}) = 3 9^{\circ}

Apply trig inverse properties

arctan (\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x}) = 3 9^{\circ}

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

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x} = tan (3 9^{\circ})

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x} = tan (3 9^{\circ})

Solve

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x} = tan (3 9^{\circ}) : x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x} = tan (3 9^{\circ})

Simplify

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x} : \frac{0.11 x}{1 - 0.001 x ^{2}}

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x}

Add similar elements:

0.1 x + 0.01 x = 0.11 x

= \frac{0.11 x}{1 - 0.1 \cdot 0.01 xx}

1 - 0.1 x \cdot 0.01 x = 1 - 0.001 x^{2}

1 - 0.1 x \cdot 0.01 x

0.1 x \cdot 0.01 x = 0.001 x^{2}

0.1 x \cdot 0.01 x

Multiply the numbers:

0.1 \cdot 0.01 = 0.001

= 0.001 xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 0.001 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 0.001 x^{2}

= 1 - 0.001 x^{2}

= \frac{0.11 x}{1 - 0.001 x ^{2}}

\frac{0.11 x}{1 - 0.001 x ^{2}} = tan (3 9^{\circ})

Multiply both sides by

1 - 0.001 x^{2}

\frac{0.11 x}{1 - 0.001 x ^{2}} = tan (3 9^{\circ})

Multiply both sides by

1 - 0.001 x^{2}

\frac{0.11 x}{1 - 0.001 x ^{2}} (1 - 0.001 x^{2}) = tan (3 9^{\circ}) (1 - 0.001 x^{2})

Simplify

0.11 x = tan (3 9^{\circ}) (1 - 0.001 x^{2})

0.11 x = tan (3 9^{\circ}) (1 - 0.001 x^{2})

Solve

0.11 x = tan (3 9^{\circ}) (1 - 0.001 x^{2}) : x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

0.11 x = tan (3 9^{\circ}) (1 - 0.001 x^{2})

Expand

tan (3 9^{\circ}) (1 - 0.001 x^{2}) : tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2}

tan (3 9^{\circ}) (1 - 0.001 x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = tan (3 9^{\circ}), b = 1, c = 0.001 x^{2}

= tan (3 9^{\circ}) \cdot 1 - tan (3 9^{\circ}) \cdot 0.001 x^{2}

= 1 \cdot tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2}

Multiply:

1 \cdot tan (3 9^{\circ}) = tan (3 9^{\circ})

= tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2}

0.11 x = tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2}

Switch sides

tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2} = 0.11 x

Move

0.11 x

to the left side

tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2} = 0.11 x

Subtract

0.11 x

from both sides

tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2} - 0.11 x = 0.11 x - 0.11 x

Simplify

tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2} - 0.11 x = 0

tan (3 9^{\circ}) - 0.001 tan (3 9^{\circ}) x^{2} - 0.11 x = 0

Write in the standard form

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

- 0.00080 \dots x^{2} - 0.11 x + 0.80978 \dots = 0

Solve with the quadratic formula

- 0.00080 \dots x^{2} - 0.11 x + 0.80978 \dots = 0

Quadratic Equation Formula:

For

a = - 0.00080 \dots, b = - 0.11, c = 0.80978 \dots

x_{1, 2} = \frac{- ( - 0.11 ) \pm ( - 0.11 ) ^{2} - 4 ( - 0.00080 \dots ) \cdot 0.80978 \dots}{2 ( - 0.00080 \dots )}

x_{1, 2} = \frac{- ( - 0.11 ) \pm ( - 0.11 ) ^{2} - 4 ( - 0.00080 \dots ) \cdot 0.80978 \dots}{2 ( - 0.00080 \dots )}

(- 0.11)^{2} - 4 (- 0.00080 \dots) \cdot 0.80978 \dots = 0.01472 \dots

(- 0.11)^{2} - 4 (- 0.00080 \dots) \cdot 0.80978 \dots

Apply rule

- (- a) = a

= (- 0.11)^{2} + 4 \cdot 0.00080 \dots \cdot 0.80978 \dots

Apply exponent rule:

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

n

is even

(- 0.11)^{2} = 0.1 1^{2}

= 0.1 1^{2} + 4 \cdot 0.00080 \dots \cdot 0.80978 \dots

Multiply the numbers:

4 \cdot 0.00080 \dots \cdot 0.80978 \dots = 0.00262 \dots

= 0.1 1^{2} + 0.00262 \dots

0.1 1^{2} = 0.0121

= 0.0121 + 0.00262 \dots

Add the numbers:

0.0121 + 0.00262 \dots = 0.01472 \dots

= 0.01472 \dots

x_{1, 2} = \frac{- ( - 0.11 ) \pm 0.01472 \dots}{2 ( - 0.00080 \dots )}

Separate the solutions

x_{1} = \frac{- ( - 0.11 ) + 0.01472 \dots}{2 ( - 0.00080 \dots )}, x_{2} = \frac{- ( - 0.11 ) - 0.01472 \dots}{2 ( - 0.00080 \dots )}

x = \frac{- ( - 0.11 ) + 0.01472 \dots}{2 ( - 0.00080 \dots )} : - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}

\frac{- ( - 0.11 ) + 0.01472 \dots}{2 ( - 0.00080 \dots )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{0.11 + 0.01472 \dots}{- 2 \cdot 0.00080 \dots}

Multiply the numbers:

2 \cdot 0.00080 \dots = 0.00161 \dots

= \frac{0.11 + 0.01472 \dots}{- 0.00161 \dots}

Apply the fraction rule:

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

= - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}

x = \frac{- ( - 0.11 ) - 0.01472 \dots}{2 ( - 0.00080 \dots )} : \frac{0.01472 \dots - 0.11}{0.00161 \dots}

\frac{- ( - 0.11 ) - 0.01472 \dots}{2 ( - 0.00080 \dots )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{0.11 - 0.01472 \dots}{- 2 \cdot 0.00080 \dots}

Multiply the numbers:

2 \cdot 0.00080 \dots = 0.00161 \dots

= \frac{0.11 - 0.01472 \dots}{- 0.00161 \dots}

Apply the fraction rule:

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

0.11 - 0.01472 \dots = - (0.01472 \dots - 0.11)

= \frac{0.01472 \dots - 0.11}{0.00161 \dots}

The solutions to the quadratic equation are:

x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

Verify Solutions

Find undefined (singularity) points:

x = 1010, x = - 1010

Take the denominator(s) of

\frac{0.1 x + 0.01 x}{1 - 0.1 x \cdot 0.01 x}

and compare to zero

Solve

1 - 0.1 x \cdot 0.01 x = 0 : x = 1010, x = - 1010

1 - 0.1 x \cdot 0.01 x = 0

Move

1

to the right side

1 - 0.1 x \cdot 0.01 x = 0

Subtract

1

from both sides

1 - 0.1 x \cdot 0.01 x - 1 = 0 - 1

Simplify

- 0.1 x \cdot 0.01 x = - 1

- 0.1 x \cdot 0.01 x = - 1

Simplify

- 0.001 x^{2} = - 1

Divide both sides by

- 0.001

\frac{- 0.001 x ^{2}}{- 0.001} = \frac{- 1}{- 0.001}

x^{2} = \frac{1}{0.001}

For

x^{2} = f (a)

the solutions are

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

x = \frac{1}{0.001}, x = - \frac{1}{0.001}

\frac{1}{0.001} = 1010

\frac{1}{0.001}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{0.001}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{0.001}

0.001 = \frac{1}{10 10}

0.001

0.001 = \frac{1}{1000}

0.001

Multiply and divide by 10 for every number after the decimal point.

There are

3

digits to the right of the decimal point, therefore multiply and divide by

1000

= \frac{1000 \cdot 0.001}{1000}

Multiply the numbers:

1000 \cdot 0.001 = 1

= \frac{1}{1000}

= \frac{1}{1000}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{1000}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{1000}

1000 = 1010

1000

Prime factorization of

1000 : 2^{3} \cdot 5^{3}

1000

1000

divides by

21000 = 500 \cdot 2

= 2 \cdot 500

500

divides by

2500 = 250 \cdot 2

= 2 \cdot 2 \cdot 250

250

divides by

2250 = 125 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

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

2, 5

are all prime numbers, therefore no further factorization is possible

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

= 2^{3} \cdot 5^{3}

= 2^{3} \cdot 5^{3}

Apply exponent rule:

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

2^{3} \cdot 5^{3} = 2^{2} \cdot 2 \cdot 5^{2} \cdot 5

= 2^{2} \cdot 2 \cdot 5^{2} \cdot 5

2^{2} \cdot 2 \cdot 5^{2} \cdot 5 = 2^{2} 5^{2} 2 \cdot 5

2^{2} \cdot 2 \cdot 5^{2} \cdot 5

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 \cdot 5^{2} \cdot 5 = 2^{2} 2 \cdot 5^{2} \cdot 5

= 2^{2} 2 \cdot 5^{2} \cdot 5

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2 \cdot 5^{2} \cdot 5 = 5^{2} 2 \cdot 5

= 2^{2} 5^{2} 2 \cdot 5

= 2^{2} 5^{2} 2 \cdot 5

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2 5^{2} 2 \cdot 5

Apply radical rule:

a^{2} = a, a \geq 0

5^{2} = 5

= 2 \cdot 5 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 1010

= \frac{1}{10 10}

= \frac{1}{\frac{1}{10 10}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{10 10}{1}

Apply the fraction rule:

\frac{a}{1} = a

= 1010

- \frac{1}{0.001} = - 1010

- \frac{1}{0.001}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{0.001}

Apply radical rule:

1 = 1

1 = 1

= - \frac{1}{0.001}

0.001 = \frac{1}{10 10}

0.001

0.001 = \frac{1}{1000}

0.001

Multiply and divide by 10 for every number after the decimal point.

There are

3

digits to the right of the decimal point, therefore multiply and divide by

1000

= \frac{1000 \cdot 0.001}{1000}

Multiply the numbers:

1000 \cdot 0.001 = 1

= \frac{1}{1000}

= \frac{1}{1000}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{1000}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{1000}

1000 = 1010

1000

Prime factorization of

1000 : 2^{3} \cdot 5^{3}

1000

1000

divides by

21000 = 500 \cdot 2

= 2 \cdot 500

500

divides by

2500 = 250 \cdot 2

= 2 \cdot 2 \cdot 250

250

divides by

2250 = 125 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

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

2, 5

are all prime numbers, therefore no further factorization is possible

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

= 2^{3} \cdot 5^{3}

= 2^{3} \cdot 5^{3}

Apply exponent rule:

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

2^{3} \cdot 5^{3} = 2^{2} \cdot 2 \cdot 5^{2} \cdot 5

= 2^{2} \cdot 2 \cdot 5^{2} \cdot 5

2^{2} \cdot 2 \cdot 5^{2} \cdot 5 = 2^{2} 5^{2} 2 \cdot 5

2^{2} \cdot 2 \cdot 5^{2} \cdot 5

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 2 \cdot 5^{2} \cdot 5 = 2^{2} 2 \cdot 5^{2} \cdot 5

= 2^{2} 2 \cdot 5^{2} \cdot 5

Apply radical rule:

ab = a b, a \geq 0, b \geq 0

2 \cdot 5^{2} \cdot 5 = 5^{2} 2 \cdot 5

= 2^{2} 5^{2} 2 \cdot 5

= 2^{2} 5^{2} 2 \cdot 5

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2 5^{2} 2 \cdot 5

Apply radical rule:

a^{2} = a, a \geq 0

5^{2} = 5

= 2 \cdot 5 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 1010

= \frac{1}{10 10}

= - \frac{1}{\frac{1}{10 10}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

\frac{1}{\frac{1}{10 10}} = \frac{10 10}{1}

= - \frac{10 10}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - 1010

x = 1010, x = - 1010

The following points are undefined

x = 1010, x = - 1010

Combine undefined points with solutions:

x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}, x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (0.1 x) + arctan (0.01 x) = 3 9^{\circ}

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

Check the solution

- \frac{0.11 + 0.01472 \dots}{0.00161 \dots} :

False

- \frac{0.11 + 0.01472 \dots}{0.00161 \dots}

Plug in

n = 1

- \frac{0.11 + 0.01472 \dots}{0.00161 \dots}

For

arctan (0.1 x) + arctan (0.01 x) = 3 9^{\circ}

plug in

x = - \frac{0.11 + 0.01472 \dots}{0.00161 \dots}

arctan (0.1 (- \frac{0.11 + 0.01472 \dots}{0.00161 \dots})) + arctan (0.01 (- \frac{0.11 + 0.01472 \dots}{0.00161 \dots})) = 3 9^{\circ}

Refine

- 2.46091 \dots = 0.68067 \dots

\Rightarrow False

Check the solution

\frac{0.01472 \dots - 0.11}{0.00161 \dots} :

True

\frac{0.01472 \dots - 0.11}{0.00161 \dots}

Plug in

n = 1

\frac{0.01472 \dots - 0.11}{0.00161 \dots}

For

arctan (0.1 x) + arctan (0.01 x) = 3 9^{\circ}

plug in

x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

arctan (0.1 \cdot \frac{0.01472 \dots - 0.11}{0.00161 \dots}) + arctan (0.01 \cdot \frac{0.01472 \dots - 0.11}{0.00161 \dots}) = 3 9^{\circ}

Refine

0.68067 \dots = 0.68067 \dots

\Rightarrow True

x = \frac{0.01472 \dots - 0.11}{0.00161 \dots}

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(0.1x)+arctan(0.01x)=39 ?
The general solution for arctan(0.1x)+arctan(0.01x)=39 is x=(sqrt(0.01472…)-0.11)/(0.00161…)