What is the general solution for sin((75pi)/(11))=sin((xpi)/(11)) ?

The general solution for sin((75pi)/(11))=sin((xpi)/(11)) is x=2+22n,x=9+22n

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sin((75pi)/(11))=sin((xpi)/(11))

Solution

sin (\frac{75 π}{11}) = sin (\frac{x π}{11})

Solution

x = 2 + 22 n, x = 9 + 22 n

Degrees

x = 114.59155 \dots^{\circ} + 1260.50714 \dots^{\circ} n, x = 515.66201 \dots^{\circ} + 1260.50714 \dots^{\circ} n

Solution steps

sin (\frac{75 π}{11}) = sin (\frac{x π}{11})

Switch sides

sin (\frac{x π}{11}) = sin (\frac{75 π}{11})

Apply trig inverse properties

sin (\frac{x π}{11}) = sin (\frac{75 π}{11})

General solutions for

sin (\frac{x π}{11}) = sin (\frac{75 π}{11})

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

\frac{x π}{11} = arcsin (sin (\frac{75 π}{11})) + 2 πn, \frac{x π}{11} = π - arcsin (sin (\frac{75 π}{11})) + 2 πn

\frac{x π}{11} = arcsin (sin (\frac{75 π}{11})) + 2 πn, \frac{x π}{11} = π - arcsin (sin (\frac{75 π}{11})) + 2 πn

Solve

\frac{x π}{11} = arcsin (sin (\frac{75 π}{11})) + 2 πn : x = 2 + 22 n

\frac{x π}{11} = arcsin (sin (\frac{75 π}{11})) + 2 πn

Simplify

arcsin (sin (\frac{75 π}{11})) + 2 πn : \frac{2 π}{11} + 2 πn

arcsin (sin (\frac{75 π}{11})) + 2 πn

arcsin (sin (\frac{75 π}{11})) = \frac{2 π}{11}

arcsin (sin (\frac{75 π}{11}))

For

- \frac{π}{2} \leq x \leq \frac{π}{2}, a rcs in (s in (x)) = x

Rewrite using trig identities:

sin (\frac{75 π}{11}) = sin (\frac{9 π}{11})

sin (\frac{75 π}{11})

sin (x + 2 π \cdot k) = sin (x)

= sin (\frac{75 π}{11} - 3 \cdot 2 π)

= sin (\frac{9 π}{11})

= arcsin (sin (\frac{9 π}{11}))

Rewrite using trig identities:

sin (\frac{75 π}{11}) = sin (\frac{2 π}{11})

sin (\frac{75 π}{11})

sin (x) = sin (π - (x))

= sin (π - \frac{9 π}{11})

= sin (\frac{2 π}{11})

= arcsin (sin (\frac{2 π}{11}))

- \frac{π}{2} \leq \frac{2 π}{11} \leq \frac{π}{2}

= \frac{2 π}{11}

= \frac{2 π}{11} + 2 πn

\frac{x π}{11} = \frac{2 π}{11} + 2 πn

Multiply both sides by

11

\frac{x π}{11} = \frac{2 π}{11} + 2 πn

Multiply both sides by

11

\frac{11 x π}{11} = 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

Simplify

\frac{11 x π}{11} = 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

Simplify

\frac{11 x π}{11} : π x

\frac{11 x π}{11}

Divide the numbers:

\frac{11}{11} = 1

= π x

Simplify

11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn : 2 π + 22 πn

11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

11 \cdot \frac{2 π}{11} = 2 π

11 \cdot \frac{2 π}{11}

Multiply fractions:

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

= \frac{2 π 11}{11}

Cancel the common factor:

11

= 2 π

11 \cdot 2 πn = 22 πn

11 \cdot 2 πn

Multiply the numbers:

11 \cdot 2 = 22

= 22 πn

= 2 π + 22 πn

π x = 2 π + 22 πn

π x = 2 π + 22 πn

π x = 2 π + 22 πn

Divide both sides by

π

π x = 2 π + 22 πn

Divide both sides by

π

\frac{π x}{π} = \frac{2 π}{π} + \frac{22 πn}{π}

Simplify

\frac{π x}{π} = \frac{2 π}{π} + \frac{22 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{2 π}{π} + \frac{22 πn}{π} : 2 + 22 n

\frac{2 π}{π} + \frac{22 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 + \frac{22 πn}{π}

Cancel

\frac{22 πn}{π} : 22 n

\frac{22 πn}{π}

Cancel the common factor:

π

= 22 n

= 2 + 22 n

x = 2 + 22 n

x = 2 + 22 n

x = 2 + 22 n

Solve

\frac{x π}{11} = π - arcsin (sin (\frac{75 π}{11})) + 2 πn : x = 9 + 22 n

\frac{x π}{11} = π - arcsin (sin (\frac{75 π}{11})) + 2 πn

Simplify

π - arcsin (sin (\frac{75 π}{11})) + 2 πn : π - \frac{2 π}{11} + 2 πn

π - arcsin (sin (\frac{75 π}{11})) + 2 πn

arcsin (sin (\frac{75 π}{11})) = \frac{2 π}{11}

arcsin (sin (\frac{75 π}{11}))

For

- \frac{π}{2} \leq x \leq \frac{π}{2}, a rcs in (s in (x)) = x

Rewrite using trig identities:

sin (\frac{75 π}{11}) = sin (\frac{9 π}{11})

sin (\frac{75 π}{11})

sin (x + 2 π \cdot k) = sin (x)

= sin (\frac{75 π}{11} - 3 \cdot 2 π)

= sin (\frac{9 π}{11})

= arcsin (sin (\frac{9 π}{11}))

Rewrite using trig identities:

sin (\frac{75 π}{11}) = sin (\frac{2 π}{11})

sin (\frac{75 π}{11})

sin (x) = sin (π - (x))

= sin (π - \frac{9 π}{11})

= sin (\frac{2 π}{11})

= arcsin (sin (\frac{2 π}{11}))

- \frac{π}{2} \leq \frac{2 π}{11} \leq \frac{π}{2}

= \frac{2 π}{11}

= π - \frac{2 π}{11} + 2 πn

\frac{x π}{11} = π - \frac{2 π}{11} + 2 πn

Multiply both sides by

11

\frac{x π}{11} = π - \frac{2 π}{11} + 2 πn

Multiply both sides by

11

\frac{11 x π}{11} = 11 π - 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

Simplify

\frac{11 x π}{11} = 11 π - 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

Simplify

\frac{11 x π}{11} : π x

\frac{11 x π}{11}

Divide the numbers:

\frac{11}{11} = 1

= π x

Simplify

11 π - 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn : 9 π + 22 πn

11 π - 11 \cdot \frac{2 π}{11} + 11 \cdot 2 πn

11 \cdot \frac{2 π}{11} = 2 π

11 \cdot \frac{2 π}{11}

Multiply fractions:

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

= \frac{2 π 11}{11}

Cancel the common factor:

11

= 2 π

11 \cdot 2 πn = 22 πn

11 \cdot 2 πn

Multiply the numbers:

11 \cdot 2 = 22

= 22 πn

= 11 π - 2 π + 22 πn

Add similar elements:

11 π - 2 π = 9 π

= 9 π + 22 πn

π x = 9 π + 22 πn

π x = 9 π + 22 πn

π x = 9 π + 22 πn

Divide both sides by

π

π x = 9 π + 22 πn

Divide both sides by

π

\frac{π x}{π} = \frac{9 π}{π} + \frac{22 πn}{π}

Simplify

\frac{π x}{π} = \frac{9 π}{π} + \frac{22 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{9 π}{π} + \frac{22 πn}{π} : 9 + 22 n

\frac{9 π}{π} + \frac{22 πn}{π}

Cancel

\frac{9 π}{π} : 9

\frac{9 π}{π}

Cancel the common factor:

π

= 9

= 9 + \frac{22 πn}{π}

Cancel

\frac{22 πn}{π} : 22 n

\frac{22 πn}{π}

Cancel the common factor:

π

= 22 n

= 9 + 22 n

x = 9 + 22 n

x = 9 + 22 n

x = 9 + 22 n

x = 2 + 22 n, x = 9 + 22 n

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Frequently Asked Questions (FAQ)

What is the general solution for sin((75pi)/(11))=sin((xpi)/(11)) ?
The general solution for sin((75pi)/(11))=sin((xpi)/(11)) is x=2+22n,x=9+22n