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Popular >

sin(x^2-2x+1)=0

Solution

sin (x^{2} - 2 x + 1) = 0

Solution

x = 1 + 2 πn, x = 1 - 2 πn, x = 1 + π (2 n + 1), x = 1 - π (2 n + 1)

Degrees

x = 57.29577 \dots^{\circ} + 143.61922 \dots^{\circ} n, x = 57.29577 \dots^{\circ} - 143.61922 \dots^{\circ} n, x = 57.29577 \dots^{\circ} + 175.89690 \dots^{\circ} n, x = 57.29577 \dots^{\circ} - 175.89690 \dots^{\circ} n

Solution steps

sin (x^{2} - 2 x + 1) = 0

General solutions for

sin (x^{2} - 2 x + 1) = 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}

x^{2} - 2 x + 1 = 0 + 2 πn, x^{2} - 2 x + 1 = π + 2 πn

x^{2} - 2 x + 1 = 0 + 2 πn, x^{2} - 2 x + 1 = π + 2 πn

Solve

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

x^{2} - 2 x + 1 = 0 + 2 πn

Expand

0 + 2 πn : 2 πn

0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

x^{2} - 2 x + 1 = 2 πn

Move

2 πn

to the left side

x^{2} - 2 x + 1 = 2 πn

Subtract

2 πn

from both sides

x^{2} - 2 x + 1 - 2 πn = 2 πn - 2 πn

Simplify

x^{2} - 2 x + 1 - 2 πn = 0

x^{2} - 2 x + 1 - 2 πn = 0

Solve with the quadratic formula

x^{2} - 2 x + 1 - 2 πn = 0

Quadratic Equation Formula:

For

a = 1, b = - 2, c = 1 - 2 πn

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

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

Simplify

(- 2)^{2} - 4 \cdot 1 \cdot (1 - 2 πn) : 22 πn

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

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

Refine

= 4 - 4 (- 2 πn + 1)

Expand

4 - 4 (1 - 2 πn) : 8 πn

4 - 4 (1 - 2 πn)

Expand

- 4 (1 - 2 πn) : - 4 + 8 πn

- 4 (1 - 2 πn)

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = 1, c = 2 πn

= - 4 \cdot 1 - (- 4) \cdot 2 πn

Apply minus-plus rules

- (- a) = a

= - 4 \cdot 1 + 4 \cdot 2 πn

Simplify

- 4 \cdot 1 + 4 \cdot 2 πn : - 4 + 8 πn

- 4 \cdot 1 + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= - 4 + 8 πn

= - 4 + 8 πn

= 4 - 4 + 8 πn

4 - 4 = 0

= 8 πn

= 8 πn

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 8 πn

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= 22 πn

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

Separate the solutions

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

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

\frac{- ( - 2 ) + 2 2 πn}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 + 2 2 πn}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 2 πn}{2}

Factor

2 + 22 πn : 2 (1 + 2 nπ)

2 + 22 πn

Rewrite as

= 2 \cdot 1 + 22 nπ

Factor out common term

2

= 2 (1 + 2 nπ)

= \frac{2 ( 1 + 2 nπ )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 2 πn

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

\frac{- ( - 2 ) - 2 2 πn}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 - 2 2 πn}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 2 πn}{2}

Factor

2 - 22 πn : 2 (1 - 2 nπ)

2 - 22 πn

Rewrite as

= 2 \cdot 1 - 22 nπ

Factor out common term

2

= 2 (1 - 2 nπ)

= \frac{2 ( 1 - 2 nπ )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 - 2 πn

The solutions to the quadratic equation are:

x = 1 + 2 πn, x = 1 - 2 πn

Solve

x^{2} - 2 x + 1 = π + 2 πn : x = 1 + π (2 n + 1), x = 1 - π (2 n + 1)

x^{2} - 2 x + 1 = π + 2 πn

Move

2 πn

to the left side

x^{2} - 2 x + 1 = π + 2 πn

Subtract

2 πn

from both sides

x^{2} - 2 x + 1 - 2 πn = π + 2 πn - 2 πn

Simplify

x^{2} - 2 x + 1 - 2 πn = π

x^{2} - 2 x + 1 - 2 πn = π

Move

π

to the left side

x^{2} - 2 x + 1 - 2 πn = π

Subtract

π

from both sides

x^{2} - 2 x + 1 - 2 πn - π = π - π

Simplify

x^{2} - 2 x + 1 - 2 πn - π = 0

x^{2} - 2 x + 1 - 2 πn - π = 0

Solve with the quadratic formula

x^{2} - 2 x + 1 - 2 πn - π = 0

Quadratic Equation Formula:

For

a = 1, b = - 2, c = 1 - 2 πn - π

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

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

Simplify

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

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

Multiply the numbers:

4 \cdot 1 = 4

= 2^{2} - 4 (- 2 πn - π + 1)

Factor

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

2^{2} - 4 (1 - 2 πn - π)

Rewrite as

= 4 \cdot 1 - 4 (1 - π - 2 nπ)

Factor out common term

4

= 4 (1 - (1 - π - 2 nπ))

Factor

- (- π - 2 πn + 1) + 1 : π (1 + 2 n)

1 - (1 - π - 2 nπ)

= 1 - (1 - π - 2 πn)

- (1 - π - 2 nπ) : - 1 + π + 2 nπ

- (1 - π - 2 nπ)

Distribute parentheses

= - (1) - (- π) - (- 2 nπ)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + π + 2 nπ

= 1 - 1 + π + 2 nπ

1 - 1 = 0

= π + 2 πn

Factor out common term

π

= π (1 + 2 n)

= 4 π (2 n + 1)

= 4 π (1 + 2 n)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 4 π (2 n + 1)

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= 2 π (2 n + 1)

x_{1, 2} = \frac{- ( - 2 ) \pm 2 π ( 1 + 2 n )}{2 \cdot 1}

Separate the solutions

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

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

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

Apply rule

- (- a) = a

= \frac{2 + 2 π ( 1 + 2 n )}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 π ( 2 n + 1 )}{2}

Factor

2 + 2 π (1 + 2 n) : 2 (1 + (1 + 2 n) π)

2 + 2 π (1 + 2 n)

Rewrite as

= 2 \cdot 1 + 2 (1 + 2 n) π

Factor out common term

2

= 2 (1 + (1 + 2 n) π)

= \frac{2 ( 1 + ( 1 + 2 n ) π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + π (2 n + 1)

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 π ( 2 n + 1 )}{2}

Factor

2 - 2 π (1 + 2 n) : 2 (1 - (1 + 2 n) π)

2 - 2 π (1 + 2 n)

Rewrite as

= 2 \cdot 1 - 2 (1 + 2 n) π

Factor out common term

2

= 2 (1 - (1 + 2 n) π)

= \frac{2 ( 1 - ( 1 + 2 n ) π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 - π (2 n + 1)

The solutions to the quadratic equation are:

x = 1 + π (2 n + 1), x = 1 - π (2 n + 1)

x = 1 + 2 πn, x = 1 - 2 πn, x = 1 + π (2 n + 1), x = 1 - π (2 n + 1)

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