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Popular >

sin(x)=sin(x^2)

Solution

sin (x) = sin (x^{2})

Solution

x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

Degrees

x = - 28.64788 \dots^{\circ} + 228.88185 \dots^{\circ} n, x = - 28.64788 \dots^{\circ} - 228.88185 \dots^{\circ} n, x = - 28.64788 \dots^{\circ} + 270.20988 \dots^{\circ} n, x = - 28.64788 \dots^{\circ} - 270.20988 \dots^{\circ} n

Solution steps

sin (x) = sin (x^{2})

Subtract

sin (x^{2})

from both sides

sin (x) - sin (x^{2}) = 0

Rewrite using trig identities

sin (x) - sin (x^{2})

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{x - x ^{2}}{2}) cos (\frac{x + x ^{2}}{2})

2 cos (\frac{x + x ^{2}}{2}) sin (\frac{x - x ^{2}}{2}) = 0

Solving each part separately

cos (\frac{x + x ^{2}}{2}) = 0 or sin (\frac{x - x ^{2}}{2}) = 0

cos (\frac{x + x ^{2}}{2}) = 0 : x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

cos (\frac{x + x ^{2}}{2}) = 0

General solutions for

cos (\frac{x + x ^{2}}{2}) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{x + x ^{2}}{2} = \frac{π}{2} + 2 πn, \frac{x + x ^{2}}{2} = \frac{3 π}{2} + 2 πn

\frac{x + x ^{2}}{2} = \frac{π}{2} + 2 πn, \frac{x + x ^{2}}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{x + x ^{2}}{2} = \frac{π}{2} + 2 πn : x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

x + x^{2} = π + 4 πn

x + x^{2} = π + 4 πn

Move

4 πn

to the left side

x + x^{2} = π + 4 πn

Subtract

4 πn

from both sides

x + x^{2} - 4 πn = π + 4 πn - 4 πn

Simplify

x + x^{2} - 4 πn = π

x + x^{2} - 4 πn = π

Move

π

to the left side

x + x^{2} - 4 πn = π

Subtract

π

from both sides

x + x^{2} - 4 πn - π = π - π

Simplify

x + x^{2} - 4 πn - π = 0

x + x^{2} - 4 πn - π = 0

Write in the standard form

a x^{2} + b x + c = 0

x^{2} + x - 4 πn - π = 0

Solve with the quadratic formula

x^{2} + x - 4 πn - π = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 4 πn - π

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 4 πn - π )}{2 \cdot 1}

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 4 πn - π )}{2 \cdot 1}

Simplify

1^{2} - 4 \cdot 1 \cdot (- 4 πn - π) : 1 - 4 (- 4 πn - π)

1^{2} - 4 \cdot 1 \cdot (- 4 πn - π)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 4 πn - π)

Multiply the numbers:

4 \cdot 1 = 4

= 1 - 4 (- 4 πn - π)

x_{1, 2} = \frac{- 1 \pm 1 - 4 ( - 4 πn - π )}{2 \cdot 1}

Separate the solutions

x_{1} = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2 \cdot 1}, x_{2} = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2 \cdot 1}

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

\frac{- 1 + 1 - 4 ( - 4 πn - π )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + - 4 ( - 4 πn - π ) + 1}{2}

x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2 \cdot 1} : \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}

\frac{- 1 - 1 - 4 ( - 4 πn - π )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - - 4 ( - 4 πn - π ) + 1}{2}

The solutions to the quadratic equation are:

x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}

Solve

\frac{x + x ^{2}}{2} = \frac{3 π}{2} + 2 πn : x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

\frac{x + x ^{2}}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x + x ^{2}}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x + x ^{2}}{2} \cdot 2 = \frac{3 π}{2} \cdot 2 + 2 πn \cdot 2

Simplify

x + x^{2} = 3 π + 4 πn

x + x^{2} = 3 π + 4 πn

Move

4 πn

to the left side

x + x^{2} = 3 π + 4 πn

Subtract

4 πn

from both sides

x + x^{2} - 4 πn = 3 π + 4 πn - 4 πn

Simplify

x + x^{2} - 4 πn = 3 π

x + x^{2} - 4 πn = 3 π

Move

3 π

to the left side

x + x^{2} - 4 πn = 3 π

Subtract

3 π

from both sides

x + x^{2} - 4 πn - 3 π = 3 π - 3 π

Simplify

x + x^{2} - 4 πn - 3 π = 0

x + x^{2} - 4 πn - 3 π = 0

Write in the standard form

a x^{2} + b x + c = 0

x^{2} + x - 4 πn - 3 π = 0

Solve with the quadratic formula

x^{2} + x - 4 πn - 3 π = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 4 πn - 3 π

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 4 πn - 3 π )}{2 \cdot 1}

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 4 πn - 3 π )}{2 \cdot 1}

Simplify

1^{2} - 4 \cdot 1 \cdot (- 4 πn - 3 π) : 1 - 4 (- 4 πn - 3 π)

1^{2} - 4 \cdot 1 \cdot (- 4 πn - 3 π)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 4 πn - 3 π)

Multiply the numbers:

4 \cdot 1 = 4

= 1 - 4 (- 4 πn - 3 π)

x_{1, 2} = \frac{- 1 \pm 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1}

Separate the solutions

x_{1} = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1}, x_{2} = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1}

x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1} : \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}

\frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + - 4 ( - 4 πn - 3 π ) + 1}{2}

x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1} : \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

\frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - - 4 ( - 4 πn - 3 π ) + 1}{2}

The solutions to the quadratic equation are:

x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

sin (\frac{x - x ^{2}}{2}) = 0 :

No Solution

sin (\frac{x - x ^{2}}{2}) = 0

General solutions for

sin (\frac{x - x ^{2}}{2}) = 0

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 ^{2}}{2} = 0 + 2 πn, \frac{x - x ^{2}}{2} = π + 2 πn

\frac{x - x ^{2}}{2} = 0 + 2 πn, \frac{x - x ^{2}}{2} = π + 2 πn

Solve

\frac{x - x ^{2}}{2} = 0 + 2 πn : x = - \frac{- 1 + 1 - 16 πn}{2}, x = - \frac{- 1 - 1 - 16 πn}{2}

\frac{x - x ^{2}}{2} = 0 + 2 πn

Multiply both sides by

2

\frac{x - x ^{2}}{2} = 0 + 2 πn

Multiply both sides by

2

\frac{x - x ^{2}}{2} \cdot 2 = 0 \cdot 2 + 2 πn \cdot 2

Simplify

x - x^{2} = 0 + 4 πn

x - x^{2} = 0 + 4 πn

x - x^{2} = 4 πn

Move

4 πn

to the left side

x - x^{2} = 4 πn

Subtract

4 πn

from both sides

x - x^{2} - 4 πn = 4 πn - 4 πn

Simplify

x - x^{2} - 4 πn = 0

x - x^{2} - 4 πn = 0

Write in the standard form

a x^{2} + b x + c = 0

- x^{2} + x - 4 πn = 0

Solve with the quadratic formula

- x^{2} + x - 4 πn = 0

Quadratic Equation Formula:

For

a = - 1, b = 1, c = - 4 πn

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) ( - 4 πn )}{2 ( - 1 )}

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) ( - 4 πn )}{2 ( - 1 )}

Simplify

1^{2} - 4 (- 1) (- 4 πn) : 1 - 16 πn

1^{2} - 4 (- 1) (- 4 πn)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) (- 4 πn)

Apply rule

- (- a) = a

= 1 - 4 \cdot 1 \cdot 4 πn

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 1 - 16 πn

x_{1, 2} = \frac{- 1 \pm 1 - 16 πn}{2 ( - 1 )}

Separate the solutions

x_{1} = \frac{- 1 + 1 - 16 πn}{2 ( - 1 )}, x_{2} = \frac{- 1 - 1 - 16 πn}{2 ( - 1 )}

x = \frac{- 1 + 1 - 16 πn}{2 ( - 1 )} : - \frac{- 1 + 1 - 16 πn}{2}

\frac{- 1 + 1 - 16 πn}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 1 - 16 πn}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + - 16 πn + 1}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 1 + 1 - 16 πn}{2}

x = \frac{- 1 - 1 - 16 πn}{2 ( - 1 )} : - \frac{- 1 - 1 - 16 πn}{2}

\frac{- 1 - 1 - 16 πn}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 1 - 16 πn}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - - 16 πn + 1}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 1 - 1 - 16 πn}{2}

The solutions to the quadratic equation are:

x = - \frac{- 1 + 1 - 16 πn}{2}, x = - \frac{- 1 - 1 - 16 πn}{2}

Solve

\frac{x - x ^{2}}{2} = π + 2 πn : x = - \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2}, x = - \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2}

\frac{x - x ^{2}}{2} = π + 2 πn

Multiply both sides by

2

\frac{x - x ^{2}}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

x - x^{2} = 2 π + 4 πn

x - x^{2} = 2 π + 4 πn

Move

4 πn

to the left side

x - x^{2} = 2 π + 4 πn

Subtract

4 πn

from both sides

x - x^{2} - 4 πn = 2 π + 4 πn - 4 πn

Simplify

x - x^{2} - 4 πn = 2 π

x - x^{2} - 4 πn = 2 π

Move

2 π

to the left side

x - x^{2} - 4 πn = 2 π

Subtract

2 π

from both sides

x - x^{2} - 4 πn - 2 π = 2 π - 2 π

Simplify

x - x^{2} - 4 πn - 2 π = 0

x - x^{2} - 4 πn - 2 π = 0

Write in the standard form

a x^{2} + b x + c = 0

- x^{2} + x - 4 πn - 2 π = 0

Solve with the quadratic formula

- x^{2} + x - 4 πn - 2 π = 0

Quadratic Equation Formula:

For

a = - 1, b = 1, c = - 4 πn - 2 π

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) ( - 4 πn - 2 π )}{2 ( - 1 )}

x_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) ( - 4 πn - 2 π )}{2 ( - 1 )}

Simplify

1^{2} - 4 (- 1) (- 4 πn - 2 π) : 1 + 4 (- 4 πn - 2 π)

1^{2} - 4 (- 1) (- 4 πn - 2 π)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) (- 4 πn - 2 π)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot (- 4 πn - 2 π)

Multiply the numbers:

4 \cdot 1 = 4

= 1 + 4 (- 4 πn - 2 π)

x_{1, 2} = \frac{- 1 \pm 1 + 4 ( - 4 πn - 2 π )}{2 ( - 1 )}

Separate the solutions

x_{1} = \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2 ( - 1 )}, x_{2} = \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2 ( - 1 )}

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

\frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + 4 ( - 4 πn - 2 π ) + 1}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2}

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

\frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 - 4 ( - 4 πn - 2 π ) + 1}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2}

The solutions to the quadratic equation are:

x = - \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2}, x = - \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2}

x = - \frac{- 1 + 1 - 16 πn}{2}, x = - \frac{- 1 - 1 - 16 πn}{2}, x = - \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2}, x = - \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2}

Since the equation is undefined for:

- \frac{- 1 + 1 - 16 πn}{2}, - \frac{- 1 - 1 - 16 πn}{2}, - \frac{- 1 + 1 + 4 ( - 4 πn - 2 π )}{2}, - \frac{- 1 - 1 + 4 ( - 4 πn - 2 π )}{2}

No Solution

Combine all the solutions

x = \frac{- 1 + 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - π )}{2}, x = \frac{- 1 + 1 - 4 ( - 4 πn - 3 π )}{2}, x = \frac{- 1 - 1 - 4 ( - 4 πn - 3 π )}{2}

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