What is the general solution for 2arctan(1/2)-arctan(x)= pi/4 ?

The general solution for 2arctan(1/2)-arctan(x)= pi/4 is x=tan((8*0.46364…-pi)/4)

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Popular Trigonometry >

2arctan(1/2)-arctan(x)= pi/4

Solution

2 arctan (\frac{1}{2}) - arctan (x) = \frac{π}{4}

Solution

x = tan (\frac{8 \cdot 0.46364 \dots - π}{4})

Solution steps

2 arctan (\frac{1}{2}) - arctan (x) = \frac{π}{4}

Move

2 arctan (\frac{1}{2})

to the right side

2 arctan (\frac{1}{2}) - arctan (x) = \frac{π}{4}

Subtract

2 arctan (\frac{1}{2})

from both sides

2 arctan (\frac{1}{2}) - arctan (x) - 2 arctan (\frac{1}{2}) = \frac{π}{4} - 2 arctan (\frac{1}{2})

Simplify

- arctan (x) = \frac{π}{4} - 2 arctan (\frac{1}{2})

- arctan (x) = \frac{π}{4} - 2 arctan (\frac{1}{2})

Divide both sides by

- 1

- arctan (x) = \frac{π}{4} - 2 arctan (\frac{1}{2})

Divide both sides by

- 1

\frac{- arctan ( x )}{- 1} = \frac{\frac{π}{4}}{- 1} - \frac{2 arctan ( \frac{1}{2} )}{- 1}

Simplify

\frac{- arctan ( x )}{- 1} = \frac{\frac{π}{4}}{- 1} - \frac{2 arctan ( \frac{1}{2} )}{- 1}

Simplify

\frac{- arctan ( x )}{- 1} : arctan (x)

\frac{- arctan ( x )}{- 1}

Apply the fraction rule:

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

= \frac{arctan ( x )}{1}

Apply rule

\frac{a}{1} = a

= arctan (x)

Simplify

\frac{\frac{π}{4}}{- 1} - \frac{2 arctan ( \frac{1}{2} )}{- 1} : \frac{8 arctan ( \frac{1}{2} ) - π}{4}

\frac{\frac{π}{4}}{- 1} - \frac{2 arctan ( \frac{1}{2} )}{- 1}

Apply rule

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

= \frac{\frac{π}{4} - 2 arctan ( \frac{1}{2} )}{- 1}

Apply the fraction rule:

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

\frac{π}{4} - 2 arctan (\frac{1}{2}) = - (2 arctan (\frac{1}{2}) - \frac{π}{4})

= \frac{2 arctan ( \frac{1}{2} ) - \frac{π}{4}}{1}

Join

2 arctan (\frac{1}{2}) - \frac{π}{4} : \frac{8 arctan ( \frac{1}{2} ) - π}{4}

2 arctan (\frac{1}{2}) - \frac{π}{4}

Convert element to fraction:

2 arctan (\frac{1}{2}) = \frac{2 arctan ( \frac{1}{2} ) 4}{4}

= \frac{2 arctan ( \frac{1}{2} ) \cdot 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 arctan ( \frac{1}{2} ) \cdot 4 - π}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8 arctan ( \frac{1}{2} ) - π}{4}

= \frac{\frac{8 a r c t a n ( \frac{1}{2} ) - π}{4}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= \frac{8 arctan ( \frac{1}{2} ) - π}{4}

arctan (x) = \frac{8 arctan ( \frac{1}{2} ) - π}{4}

arctan (x) = \frac{8 arctan ( \frac{1}{2} ) - π}{4}

arctan (x) = \frac{8 arctan ( \frac{1}{2} ) - π}{4}

Apply trig inverse properties

arctan (x) = \frac{8 arctan ( \frac{1}{2} ) - π}{4}

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

x = tan (\frac{8 arctan ( \frac{1}{2} ) - π}{4})

x = tan (\frac{8 arctan ( \frac{1}{2} ) - π}{4})

Show solutions in decimal form

x = tan (\frac{8 \cdot 0.46364 \dots - π}{4})

Graph

Popular Examples

cos(x)=(2-tan(x))(1+sin(x))

Frequently Asked Questions (FAQ)

What is the general solution for 2arctan(1/2)-arctan(x)= pi/4 ?
The general solution for 2arctan(1/2)-arctan(x)= pi/4 is x=tan((8*0.46364…-pi)/4)