What is the general solution for 1=1-2arctan((1-2-C)/(1-C)) ?

The general solution for 1=1-2arctan((1-2-C)/(1-C)) is C=-1

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1=1-2arctan((1-2-C)/(1-C))

Solution

1 = 1 - 2 arctan (\frac{1 - 2 - C}{1 - C})

Solution

C = - 1

Solution steps

1 = 1 - 2 arctan (\frac{1 - 2 - C}{1 - C})

Switch sides

1 - 2 arctan (\frac{1 - 2 - C}{1 - C}) = 1

Subtract

1

from both sides

1 - 2 arctan (\frac{1 - 2 - C}{1 - C}) - 1 = 1 - 1

Simplify

- 2 arctan (\frac{1 - 2 - C}{1 - C}) = 0

Divide both sides by

- 2

- 2 arctan (\frac{1 - 2 - C}{1 - C}) = 0

Divide both sides by

- 2

\frac{- 2 arctan ( \frac{1 - 2 - C}{1 - C} )}{- 2} = \frac{0}{- 2}

Simplify

arctan (\frac{1 - 2 - C}{1 - C}) = 0

arctan (\frac{1 - 2 - C}{1 - C}) = 0

Apply trig inverse properties

arctan (\frac{1 - 2 - C}{1 - C}) = 0

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

\frac{1 - 2 - C}{1 - C} = 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{1 - 2 - C}{1 - C} = 0

\frac{1 - 2 - C}{1 - C} = 0

Solve

\frac{1 - 2 - C}{1 - C} = 0 : C = - 1

\frac{1 - 2 - C}{1 - C} = 0

Simplify

\frac{1 - 2 - C}{1 - C} : \frac{- C - 1}{1 - C}

\frac{1 - 2 - C}{1 - C}

Subtract the numbers:

1 - 2 = - 1

= \frac{- C - 1}{1 - C}

\frac{- C - 1}{1 - C} = 0

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

- C - 1 = 0

Move

1

to the right side

- C - 1 = 0

Add

1

to both sides

- C - 1 + 1 = 0 + 1

Simplify

- C = 1

- C = 1

Divide both sides by

- 1

- C = 1

Divide both sides by

- 1

\frac{- C}{- 1} = \frac{1}{- 1}

Simplify

C = - 1

C = - 1

Verify Solutions

Find undefined (singularity) points:

C = 1

Take the denominator(s) of

\frac{1 - 2 - C}{1 - C}

and compare to zero

Solve

1 - C = 0 : C = 1

1 - C = 0

Move

1

to the right side

1 - C = 0

Subtract

1

from both sides

1 - C - 1 = 0 - 1

Simplify

- C = - 1

- C = - 1

Divide both sides by

- 1

- C = - 1

Divide both sides by

- 1

\frac{- C}{- 1} = \frac{- 1}{- 1}

Simplify

C = 1

C = 1

The following points are undefined

C = 1

Combine undefined points with solutions:

C = - 1

C = - 1

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 1=1-2arctan((1-2-C)/(1-C)) ?
The general solution for 1=1-2arctan((1-2-C)/(1-C)) is C=-1