What is the general solution for solvefor x,y= 1/pi arctan(x/s)+1/2 ?

The general solution for solvefor x,y= 1/pi arctan(x/s)+1/2 is x=tan((2piy-pi)/2)s

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solvefor x,y= 1/pi arctan(x/s)+1/2

Solution

solve for

x, y = \frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2}

Solution

x = tan (\frac{2 π y - π}{2}) s

Solution steps

y = \frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2}

Switch sides

\frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2} = y

Solve by substitution

\frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2} = y

Let:

arctan (\frac{x}{s}) = u

\frac{1}{π} u + \frac{1}{2} = y

\frac{1}{π} u + \frac{1}{2} = y : u = π y - \frac{π}{2}

\frac{1}{π} u + \frac{1}{2} = y

Move

\frac{1}{2}

to the right side

\frac{1}{π} u + \frac{1}{2} = y

Subtract

\frac{1}{2}

from both sides

\frac{1}{π} u + \frac{1}{2} - \frac{1}{2} = y - \frac{1}{2}

Simplify

\frac{1}{π} u = y - \frac{1}{2}

\frac{1}{π} u = y - \frac{1}{2}

Multiply both sides by

π

\frac{1}{π} u = y - \frac{1}{2}

Multiply both sides by

π

\frac{1}{π} u π = y π - \frac{1}{2} π

Simplify

\frac{1}{π} u π = y π - \frac{1}{2} π

Simplify

\frac{1}{π} u π : u

\frac{1}{π} u π

Multiply fractions:

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

= \frac{1 π}{π} u

Cancel the common factor:

π

= u \cdot 1

Multiply:

u \cdot 1 = u

= u

Simplify

y π - \frac{1}{2} π : π y - \frac{π}{2}

y π - \frac{1}{2} π

\frac{1}{2} π = \frac{π}{2}

\frac{1}{2} π

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

= π y - \frac{π}{2}

u = π y - \frac{π}{2}

u = π y - \frac{π}{2}

u = π y - \frac{π}{2}

arctan (\frac{x}{s}) = π y - \frac{π}{2}

Substitute back

u = arctan (\frac{x}{s})

arctan (\frac{x}{s}) = π y - \frac{π}{2}

arctan (\frac{x}{s}) = π y - \frac{π}{2}

Subtract

y

from both sides

\frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2} - y = 0

Simplify

\frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2} - y : \frac{2 arctan ( \frac{x}{s} ) + π - 2 π y}{2 π}

\frac{1}{π} arctan (\frac{x}{s}) + \frac{1}{2} - y

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

\frac{1}{π} arctan (\frac{x}{s})

Multiply fractions:

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

= \frac{1 \cdot arctan ( \frac{x}{s} )}{π}

Multiply:

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

= \frac{arctan ( \frac{x}{s} )}{π}

= \frac{arctan ( \frac{x}{s} )}{π} + \frac{1}{2} - y

Convert element to fraction:

y = \frac{y}{1}

= \frac{arctan ( \frac{x}{s} )}{π} + \frac{1}{2} - \frac{y}{1}

Least Common Multiplier of

π, 2, 1 : 2 π

π, 2, 1

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 1 : 2

2, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

1

Multiply each factor the greatest number of times it occurs in either

2

1

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 π

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 π

For

\frac{arctan ( \frac{x}{s} )}{π} :

multiply the denominator and numerator by

2

\frac{arctan ( \frac{x}{s} )}{π} = \frac{arctan ( \frac{x}{s} ) \cdot 2}{π 2}

For

\frac{1}{2} :

multiply the denominator and numerator by

π

\frac{1}{2} = \frac{1 π}{2 π} = \frac{π}{2 π}

For

\frac{y}{1} :

multiply the denominator and numerator by

2 π

\frac{y}{1} = \frac{y \cdot 2 π}{1 \cdot 2 π} = \frac{y \cdot 2 π}{2 π}

= \frac{arctan ( \frac{x}{s} ) \cdot 2}{π 2} + \frac{π}{2 π} - \frac{y \cdot 2 π}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{arctan ( \frac{x}{s} ) \cdot 2 + π - y \cdot 2 π}{2 π}

\frac{2 arctan ( \frac{x}{s} ) + π - 2 π y}{2 π} = 0

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

2 arctan (\frac{x}{s}) + π - 2 π y = 0

Solve by substitution

2 arctan (\frac{x}{s}) + π - 2 π y = 0

Let:

arctan (\frac{x}{s}) = u

2 u + π - 2 π y = 0

2 u + π - 2 π y = 0 : u = \frac{2 π y - π}{2}

2 u + π - 2 π y = 0

Move

2 π y

to the right side

2 u + π - 2 π y = 0

Add

2 π y

to both sides

2 u + π - 2 π y + 2 π y = 0 + 2 π y

Simplify

2 u + π = 2 π y

2 u + π = 2 π y

Move

π

to the right side

2 u + π = 2 π y

Subtract

π

from both sides

2 u + π - π = 2 π y - π

Simplify

2 u = 2 π y - π

2 u = 2 π y - π

Divide both sides by

2

2 u = 2 π y - π

Divide both sides by

2

\frac{2 u}{2} = \frac{2 π y}{2} - \frac{π}{2}

Simplify

\frac{2 u}{2} = \frac{2 π y}{2} - \frac{π}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Divide the numbers:

\frac{2}{2} = 1

= u

Simplify

\frac{2 π y}{2} - \frac{π}{2} : \frac{2 π y - π}{2}

\frac{2 π y}{2} - \frac{π}{2}

Apply rule

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

= \frac{2 π y - π}{2}

u = \frac{2 π y - π}{2}

u = \frac{2 π y - π}{2}

u = \frac{2 π y - π}{2}

arctan (\frac{x}{s}) = \frac{2 π y - π}{2}

Substitute back

u = arctan (\frac{x}{s})

arctan (\frac{x}{s}) = \frac{2 π y - π}{2}

arctan (\frac{x}{s}) = \frac{2 π y - π}{2}

Let:

u = \frac{x}{s}

2 arctan (u) + π - 2 π y = 0

Move

2 π y

to the right side

2 arctan (u) + π - 2 π y = 0

Add

2 π y

to both sides

2 arctan (u) + π - 2 π y + 2 π y = 0 + 2 π y

Simplify

2 arctan (u) + π = 2 π y

2 arctan (u) + π = 2 π y

Move

π

to the right side

2 arctan (u) + π = 2 π y

Subtract

π

from both sides

2 arctan (u) + π - π = 2 π y - π

Simplify

2 arctan (u) = 2 π y - π

2 arctan (u) = 2 π y - π

Divide both sides by

2

2 arctan (u) = 2 π y - π

Divide both sides by

2

\frac{2 arctan ( u )}{2} = \frac{2 π y}{2} - \frac{π}{2}

Simplify

\frac{2 arctan ( u )}{2} = \frac{2 π y}{2} - \frac{π}{2}

Simplify

\frac{2 arctan ( u )}{2} : arctan (u)

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

Divide the numbers:

\frac{2}{2} = 1

= arctan (u)

Simplify

\frac{2 π y}{2} - \frac{π}{2} : \frac{2 π y - π}{2}

\frac{2 π y}{2} - \frac{π}{2}

Apply rule

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

= \frac{2 π y - π}{2}

arctan (u) = \frac{2 π y - π}{2}

arctan (u) = \frac{2 π y - π}{2}

arctan (u) = \frac{2 π y - π}{2}

Apply trig inverse properties

arctan (u) = \frac{2 π y - π}{2}

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

u = tan (\frac{2 π y - π}{2})

u = tan (\frac{2 π y - π}{2})

Substitute back

u = \frac{x}{s}

\frac{x}{s} = tan (\frac{2 π y - π}{2}) : x = tan (\frac{2 π y - π}{2}) s; s \neq = 0

\frac{x}{s} = tan (\frac{2 π y - π}{2})

Multiply both sides by

s

\frac{x}{s} = tan (\frac{2 π y - π}{2})

Multiply both sides by

s

\frac{x s}{s} = tan (\frac{2 π y - π}{2}) s; s \neq = 0

Simplify

x = tan (\frac{2 π y - π}{2}) s; s \neq = 0

x = tan (\frac{2 π y - π}{2}) s; s \neq = 0

x = tan (\frac{2 π y - π}{2}) s

Popular Examples

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sin (x) + sin (2 x) = 0, 0 \leq x \leq 2 π

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8 sin (2 x) = 16 cos (x)

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,y= 1/pi arctan(x/s)+1/2 ?
The general solution for solvefor x,y= 1/pi arctan(x/s)+1/2 is x=tan((2piy-pi)/2)s