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Popular Trigonometry >

solvefor x,sin(pi/x)=y-1

Solution

solvefor

Solution

x = \frac{π}{arcsin ( - 1 + y ) + 2 πn}, x = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

Solution steps

sin (\frac{π}{x}) = y - 1

Apply trig inverse properties

sin (\frac{π}{x}) = y - 1

General solutions for

sin (\frac{π}{x}) = y - 1

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

\frac{π}{x} = arcsin (y - 1) + 2 πn, \frac{π}{x} = π + arcsin (- y + 1) + 2 πn

\frac{π}{x} = arcsin (y - 1) + 2 πn, \frac{π}{x} = π + arcsin (- y + 1) + 2 πn

Solve

\frac{π}{x} = arcsin (y - 1) + 2 πn : x = \frac{π}{arcsin ( - 1 + y ) + 2 πn}

\frac{π}{x} = arcsin (y - 1) + 2 πn

Multiply both sides by

x

\frac{π}{x} = arcsin (y - 1) + 2 πn

Multiply both sides by

x

\frac{π}{x} x = arcsin (y - 1) x + 2 πn x

Simplify

π = arcsin (y - 1) x + 2 πn x

π = arcsin (y - 1) x + 2 πn x

Switch sides

arcsin (y - 1) x + 2 πn x = π

Factor

arcsin (y - 1) x + 2 πn x : x (arcsin (- 1 + y) + 2 πn)

arcsin (y - 1) x + 2 πn x

Factor out common term

x

= x (arcsin (- 1 + y) + 2 πn)

x (arcsin (- 1 + y) + 2 πn) = π

Divide both sides by

arcsin (- 1 + y) + 2 πn

x (arcsin (- 1 + y) + 2 πn) = π

Divide both sides by

arcsin (- 1 + y) + 2 πn

\frac{x ( arcsin ( - 1 + y ) + 2 πn )}{arcsin ( - 1 + y ) + 2 πn} = \frac{π}{arcsin ( - 1 + y ) + 2 πn}

Simplify

x = \frac{π}{arcsin ( - 1 + y ) + 2 πn}

x = \frac{π}{arcsin ( - 1 + y ) + 2 πn}

Solve

\frac{π}{x} = π + arcsin (- y + 1) + 2 πn : x = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

\frac{π}{x} = π + arcsin (- y + 1) + 2 πn

Multiply both sides by

x

\frac{π}{x} = π + arcsin (- y + 1) + 2 πn

Multiply both sides by

x

\frac{π}{x} x = π x + arcsin (- y + 1) x + 2 πn x

Simplify

π = π x + arcsin (- y + 1) x + 2 πn x

π = π x + arcsin (- y + 1) x + 2 πn x

Switch sides

π x + arcsin (- y + 1) x + 2 πn x = π

Factor

π x + arcsin (- y + 1) x + 2 πn x : x (π + arcsin (1 - y) + 2 πn)

π x + arcsin (- y + 1) x + 2 πn x

Factor out common term

x

= x (π + arcsin (1 - y) + 2 πn)

x (π + arcsin (1 - y) + 2 πn) = π

Divide both sides by

π + arcsin (1 - y) + 2 πn

x (π + arcsin (1 - y) + 2 πn) = π

Divide both sides by

π + arcsin (1 - y) + 2 πn

\frac{x ( π + arcsin ( 1 - y ) + 2 πn )}{π + arcsin ( 1 - y ) + 2 πn} = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

Simplify

x = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

x = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

x = \frac{π}{arcsin ( - 1 + y ) + 2 πn}, x = \frac{π}{π + arcsin ( 1 - y ) + 2 πn}

Graph

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