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

The general solution for sin(1/x)=1 is x= 2/(pi(1+4n));\quad n\ne-1/4

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sin(1/x)=1

Solution

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

Solution

x = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

Degrees

x = 0^{\circ} + 7.29512 \dots^{\circ} n

Solution steps

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

General solutions for

sin (\frac{1}{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{1}{x} = \frac{π}{2} + 2 πn

\frac{1}{x} = \frac{π}{2} + 2 πn

Solve

\frac{1}{x} = \frac{π}{2} + 2 πn : x = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

\frac{1}{x} = \frac{π}{2} + 2 πn

Multiply both sides by

x

\frac{1}{x} = \frac{π}{2} + 2 πn

Multiply both sides by

x

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

Simplify

1 = \frac{π}{2} x + 2 πn x

1 = \frac{π}{2} x + 2 πn x

Switch sides

\frac{π}{2} x + 2 πn x = 1

Multiply both sides by

2

\frac{π}{2} x + 2 πn x = 1

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{π}{2} x \cdot 2 : π x

\frac{π}{2} x \cdot 2

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

2 πn x \cdot 2 : 4 πn x

2 πn x \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn x

Simplify

1 \cdot 2 : 2

1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

π x + 4 πn x = 2

π x + 4 πn x = 2

π x + 4 πn x = 2

Factor

π x + 4 πn x : π x (1 + 4 n)

π x + 4 πn x

Factor out common term

x π

= x π (1 + 4 n)

π x (1 + 4 n) = 2

Divide both sides by

π (1 + 4 n); n \neq = - \frac{1}{4}

π x (1 + 4 n) = 2

Divide both sides by

π (1 + 4 n); n \neq = - \frac{1}{4}

\frac{π x ( 1 + 4 n )}{π ( 1 + 4 n )} = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

Simplify

x = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

x = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

x = \frac{2}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

Popular Examples

2sin(θ)-cos(θ)=0

2 sin (θ) - cos (θ) = 0

cot(x/7)=sqrt(12)

cot (\frac{x}{7}) = 12

sin(θ)= 4/9

sin (θ) = \frac{4}{9}

solvefor y,x=sec(y)solve for

y, x = sec (y)

2cos^2(θ)-cos(θ)-4=-6cos(θ)-6

2 cos^{2} (θ) - cos (θ) - 4 = - 6 cos (θ) - 6

Frequently Asked Questions (FAQ)

What is the general solution for sin(1/x)=1 ?
The general solution for sin(1/x)=1 is x= 2/(pi(1+4n));\quad n\ne-1/4