What is the general solution for solvefor x,sin(x/x)=1 ?

The general solution for solvefor x,sin(x/x)=1 is No Solution for x\in\mathbb{R}

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Popular >

solvefor x,sin(x/x)=1

Solution

solve for

x, sin (\frac{x}{x}) = 1

Solution

No Solution for x \in R

Solution steps

sin (\frac{x}{x}) = 1

General solutions for

sin (\frac{x}{x}) = 1

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{x} = \frac{π}{2} + 2 πn

\frac{x}{x} = \frac{π}{2} + 2 πn

Solve

\frac{x}{x} = \frac{π}{2} + 2 πn : n = \frac{1}{2 π} - \frac{1}{4}

\frac{x}{x} = \frac{π}{2} + 2 πn

Simplify

\frac{x}{x} : 1

\frac{x}{x}

Apply rule

\frac{a}{a} = 1

= 1

1 = \frac{π}{2} + 2 πn

Multiply both sides by

2

1 = \frac{π}{2} + 2 πn

Multiply both sides by

2

1 \cdot 2 = \frac{π}{2} \cdot 2 + 2 πn \cdot 2

Simplify

1 \cdot 2 = \frac{π}{2} \cdot 2 + 2 πn \cdot 2

Simplify

1 \cdot 2 : 2

1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

Simplify

\frac{π}{2} \cdot 2 : π

\frac{π}{2} \cdot 2

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 = π + 4 πn

2 = π + 4 πn

2 = π + 4 πn

Switch sides

π + 4 πn = 2

Move

π

to the right side

π + 4 πn = 2

Subtract

π

from both sides

π + 4 πn - π = 2 - π

Simplify

4 πn = 2 - π

4 πn = 2 - π

Divide both sides by

4 π

4 πn = 2 - π

Divide both sides by

4 π

\frac{4 πn}{4 π} = \frac{2}{4 π} - \frac{π}{4 π}

Simplify

\frac{4 πn}{4 π} = \frac{2}{4 π} - \frac{π}{4 π}

Simplify

\frac{4 πn}{4 π} : n

\frac{4 πn}{4 π}

Divide the numbers:

\frac{4}{4} = 1

= \frac{πn}{π}

Cancel the common factor:

π

= n

Simplify

\frac{2}{4 π} - \frac{π}{4 π} : \frac{1}{2 π} - \frac{1}{4}

\frac{2}{4 π} - \frac{π}{4 π}

Cancel

\frac{2}{4 π} : \frac{1}{2 π}

\frac{2}{4 π}

Cancel the common factor:

2

= \frac{1}{2 π}

= \frac{1}{2 π} - \frac{π}{4 π}

Cancel

\frac{π}{4 π} : \frac{1}{4}

\frac{π}{4 π}

Cancel the common factor:

π

= \frac{1}{4}

= \frac{1}{2 π} - \frac{1}{4}

n = \frac{1}{2 π} - \frac{1}{4}

n = \frac{1}{2 π} - \frac{1}{4}

n = \frac{1}{2 π} - \frac{1}{4}

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 π} - \frac{1}{4}

x = \frac{1}{2 π} - \frac{1}{4}

Since the equation is undefined for:

\frac{1}{2 π} - \frac{1}{4}

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)=1 ?
The general solution for solvefor x,sin(x/x)=1 is No Solution for x\in\mathbb{R}