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Popular >

arctan(100x)-arctan(x)=45

Solution

arctan (100 x) - arctan (x) = 4 5^{\circ}

Solution

x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

Solution steps

arctan (100 x) - arctan (x) = 4 5^{\circ}

Rewrite using trig identities

arctan (100 x) - arctan (x)

Use the Sum to Product identity:

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

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

arctan (\frac{100 x - x}{1 + 100 xx}) = 4 5^{\circ}

Apply trig inverse properties

arctan (\frac{100 x - x}{1 + 100 xx}) = 4 5^{\circ}

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

\frac{100 x - x}{1 + 100 xx} = tan (4 5^{\circ})

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

Use the following trivial identity:

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

= 1

\frac{100 x - x}{1 + 100 xx} = 1

\frac{100 x - x}{1 + 100 xx} = 1

Solve

\frac{100 x - x}{1 + 100 xx} = 1 : x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

\frac{100 x - x}{1 + 100 xx} = 1

Simplify

\frac{100 x - x}{1 + 100 xx} : \frac{99 x}{1 + 100 x ^{2}}

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

Add similar elements:

100 x - x = 99 x

= \frac{99 x}{1 + 100 xx}

1 + 100 xx = 1 + 100 x^{2}

1 + 100 xx

100 xx = 100 x^{2}

100 xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 100 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 100 x^{2}

= 1 + 100 x^{2}

= \frac{99 x}{1 + 100 x ^{2}}

\frac{99 x}{1 + 100 x ^{2}} = 1

Multiply both sides by

1 + 100 x^{2}

\frac{99 x}{1 + 100 x ^{2}} = 1

Multiply both sides by

1 + 100 x^{2}

\frac{99 x}{1 + 100 x ^{2}} (1 + 100 x^{2}) = 1 \cdot (1 + 100 x^{2})

Simplify

\frac{99 x}{1 + 100 x ^{2}} (1 + 100 x^{2}) = 1 \cdot (1 + 100 x^{2})

Simplify

\frac{99 x}{1 + 100 x ^{2}} (1 + 100 x^{2}) : 99 x

\frac{99 x}{1 + 100 x ^{2}} (1 + 100 x^{2})

Multiply fractions:

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

= \frac{99 x ( 1 + 100 x ^{2} )}{1 + 100 x ^{2}}

Cancel the common factor:

1 + 100 x^{2}

= 99 x

Simplify

1 \cdot (1 + 100 x^{2}) : 1 + 100 x^{2}

1 \cdot (1 + 100 x^{2})

Multiply:

1 \cdot (1 + 100 x^{2}) = (1 + 100 x^{2})

= (1 + 100 x^{2})

Remove parentheses:

(a) = a

= 1 + 100 x^{2}

99 x = 1 + 100 x^{2}

99 x = 1 + 100 x^{2}

99 x = 1 + 100 x^{2}

Solve

99 x = 1 + 100 x^{2} : x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

99 x = 1 + 100 x^{2}

Switch sides

1 + 100 x^{2} = 99 x

Move

99 x

to the left side

1 + 100 x^{2} = 99 x

Subtract

99 x

from both sides

1 + 100 x^{2} - 99 x = 99 x - 99 x

Simplify

1 + 100 x^{2} - 99 x = 0

1 + 100 x^{2} - 99 x = 0

Write in the standard form

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

100 x^{2} - 99 x + 1 = 0

Solve with the quadratic formula

100 x^{2} - 99 x + 1 = 0

Quadratic Equation Formula:

For

a = 100, b = - 99, c = 1

x_{1, 2} = \frac{- ( - 99 ) \pm ( - 99 ) ^{2} - 4 \cdot 100 \cdot 1}{2 \cdot 100}

x_{1, 2} = \frac{- ( - 99 ) \pm ( - 99 ) ^{2} - 4 \cdot 100 \cdot 1}{2 \cdot 100}

(- 99)^{2} - 4 \cdot 100 \cdot 1 = 9401

(- 99)^{2} - 4 \cdot 100 \cdot 1

Apply exponent rule:

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

n

is even

(- 99)^{2} = 9 9^{2}

= 9 9^{2} - 4 \cdot 100 \cdot 1

Multiply the numbers:

4 \cdot 100 \cdot 1 = 400

= 9 9^{2} - 400

9 9^{2} = 9801

= 9801 - 400

Subtract the numbers:

9801 - 400 = 9401

= 9401

x_{1, 2} = \frac{- ( - 99 ) \pm 9401}{2 \cdot 100}

Separate the solutions

x_{1} = \frac{- ( - 99 ) + 9401}{2 \cdot 100}, x_{2} = \frac{- ( - 99 ) - 9401}{2 \cdot 100}

x = \frac{- ( - 99 ) + 9401}{2 \cdot 100} : \frac{99 + 9401}{200}

\frac{- ( - 99 ) + 9401}{2 \cdot 100}

Apply rule

- (- a) = a

= \frac{99 + 9401}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{99 + 9401}{200}

x = \frac{- ( - 99 ) - 9401}{2 \cdot 100} : \frac{99 - 9401}{200}

\frac{- ( - 99 ) - 9401}{2 \cdot 100}

Apply rule

- (- a) = a

= \frac{99 - 9401}{2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{99 - 9401}{200}

The solutions to the quadratic equation are:

x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (100 x) - arctan (x) = 4 5^{\circ}

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

Check the solution

\frac{99 + 9401}{200} :

True

\frac{99 + 9401}{200}

Plug in

n = 1

\frac{99 + 9401}{200}

For

arctan (100 x) - arctan (x) = 4 5^{\circ}

plug in

x = \frac{99 + 9401}{200}

arctan (100 \cdot \frac{99 + 9401}{200}) - arctan (\frac{99 + 9401}{200}) = 4 5^{\circ}

Refine

0.78539 \dots = 0.78539 \dots

\Rightarrow True

Check the solution

\frac{99 - 9401}{200} :

True

\frac{99 - 9401}{200}

Plug in

n = 1

\frac{99 - 9401}{200}

For

arctan (100 x) - arctan (x) = 4 5^{\circ}

plug in

x = \frac{99 - 9401}{200}

arctan (100 \cdot \frac{99 - 9401}{200}) - arctan (\frac{99 - 9401}{200}) = 4 5^{\circ}

Refine

0.78539 \dots = 0.78539 \dots

\Rightarrow True

x = \frac{99 + 9401}{200}, x = \frac{99 - 9401}{200}

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