What is the general solution for solvefor x,sin(x/x)=0.7 ?

The general solution for solvefor x,sin(x/x)=0.7 is No Solution for x\in\mathbb{R}

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solvefor x,sin(x/x)=0.7

Solution

solve for

x, sin (\frac{x}{x}) = 0.7

Solution

No Solution for x \in R

Solution steps

sin (\frac{x}{x}) = 0.7

Apply trig inverse properties

sin (\frac{x}{x}) = 0.7

General solutions for

sin (\frac{x}{x}) = 0.7

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

\frac{x}{x} = arcsin (0.7) + 2 πn, \frac{x}{x} = π - arcsin (0.7) + 2 πn

\frac{x}{x} = arcsin (0.7) + 2 πn, \frac{x}{x} = π - arcsin (0.7) + 2 πn

Solve

\frac{x}{x} = arcsin (0.7) + 2 πn : n = \frac{1 - arcsin ( 0.7 )}{2 π}

\frac{x}{x} = arcsin (0.7) + 2 πn

Simplify

\frac{x}{x} : 1

\frac{x}{x}

Apply rule

\frac{a}{a} = 1

= 1

1 = arcsin (0.7) + 2 πn

Switch sides

arcsin (0.7) + 2 πn = 1

Move

arcsin (0.7)

to the right side

arcsin (0.7) + 2 πn = 1

Subtract

arcsin (0.7)

from both sides

arcsin (0.7) + 2 πn - arcsin (0.7) = 1 - arcsin (0.7)

Simplify

2 πn = 1 - arcsin (0.7)

2 πn = 1 - arcsin (0.7)

Divide both sides by

2 π

2 πn = 1 - arcsin (0.7)

Divide both sides by

2 π

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

Simplify

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

Simplify

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{πn}{π}

Cancel the common factor:

π

= n

Simplify

\frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π} : \frac{1 - arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

Verify Solutions

Find undefined (singularity) points:

x = 0

Take the denominator(s) of

\frac{x}{x}

and compare to zero

x = 0

The following points are undefined

x = 0

Combine undefined points with solutions:

n = \frac{1 - arcsin ( 0.7 )}{2 π}

Solve

\frac{x}{x} = π - arcsin (0.7) + 2 πn : n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{x}{x} = π - arcsin (0.7) + 2 πn

Simplify

\frac{x}{x} : 1

\frac{x}{x}

Apply rule

\frac{a}{a} = 1

= 1

1 = π - arcsin (0.7) + 2 πn

Switch sides

π - arcsin (0.7) + 2 πn = 1

Move

π

to the right side

π - arcsin (0.7) + 2 πn = 1

Subtract

π

from both sides

π - arcsin (0.7) + 2 πn - π = 1 - π

Simplify

- arcsin (0.7) + 2 πn = 1 - π

- arcsin (0.7) + 2 πn = 1 - π

Move

arcsin (0.7)

to the right side

- arcsin (0.7) + 2 πn = 1 - π

Add

arcsin (0.7)

to both sides

- arcsin (0.7) + 2 πn + arcsin (0.7) = 1 - π + arcsin (0.7)

Simplify

2 πn = 1 - π + arcsin (0.7)

2 πn = 1 - π + arcsin (0.7)

Divide both sides by

2 π

2 πn = 1 - π + arcsin (0.7)

Divide both sides by

2 π

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

Simplify

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

Simplify

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{πn}{π}

Cancel the common factor:

π

= n

Simplify

\frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π} : - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

Cancel

\frac{π}{2 π} : \frac{1}{2}

\frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

= \frac{1}{2 π} - \frac{1}{2} + \frac{arcsin ( 0.7 )}{2 π}

Combine the fractions

\frac{1}{2 π} + \frac{arcsin ( 0.7 )}{2 π} : \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + arcsin ( 0.7 )}{2 π}

= - \frac{1}{2} + \frac{arcsin ( 0.7 ) + 1}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

Verify Solutions

Find undefined (singularity) points:

x = 0

Take the denominator(s) of

\frac{x}{x}

and compare to zero

x = 0

The following points are undefined

x = 0

Combine undefined points with solutions:

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

x = \frac{1 - arcsin ( 0.7 )}{2 π}, x = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

Since the equation is undefined for:

\frac{1 - arcsin ( 0.7 )}{2 π}, - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,sin(x/x)=0.7 ?
The general solution for solvefor x,sin(x/x)=0.7 is No Solution for x\in\mathbb{R}