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Popular >

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

Solution

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

Solution

x = \frac{7 π + 12 πn}{3}, x = - \frac{7 π + 12 πn}{3}, x = \frac{11 π + 12 πn}{3}, x = - \frac{11 π + 12 πn}{3}

Degrees

x = 0^{\circ} + 255.57227 \dots^{\circ} n, x = 0^{\circ} - 255.57227 \dots^{\circ} n, x = 0^{\circ} + 281.19063 \dots^{\circ} n, x = 0^{\circ} - 281.19063 \dots^{\circ} n

Solution steps

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

Switch sides

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

2 sin (\frac{x ^{2}}{2}) = - 1

2 sin (\frac{x ^{2}}{2}) = - 1

Divide both sides by

2

2 sin (\frac{x ^{2}}{2}) = - 1

Divide both sides by

2

\frac{2 sin ( \frac{x ^{2}}{2} )}{2} = \frac{- 1}{2}

Simplify

sin (\frac{x ^{2}}{2}) = - \frac{1}{2}

sin (\frac{x ^{2}}{2}) = - \frac{1}{2}

General solutions for

sin (\frac{x ^{2}}{2}) = - \frac{1}{2}

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 ^{2}}{2} = \frac{7 π}{6} + 2 πn, \frac{x ^{2}}{2} = \frac{11 π}{6} + 2 πn

\frac{x ^{2}}{2} = \frac{7 π}{6} + 2 πn, \frac{x ^{2}}{2} = \frac{11 π}{6} + 2 πn

Solve

\frac{x ^{2}}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π + 12 πn}{3}, x = - \frac{7 π + 12 πn}{3}

\frac{x ^{2}}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{x ^{2}}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x ^{2}}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x ^{2}}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x ^{2}}{2} : x^{2}

\frac{2 x ^{2}}{2}

Divide the numbers:

\frac{2}{2} = 1

= x^{2}

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x^{2} = \frac{7 π}{3} + 4 πn

x^{2} = \frac{7 π}{3} + 4 πn

x^{2} = \frac{7 π}{3} + 4 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{7 π}{3} + 4 πn, x = - \frac{7 π}{3} + 4 πn

Simplify

\frac{7 π}{3} + 4 πn : \frac{7 π + 12 πn}{3}

\frac{7 π}{3} + 4 πn

Join

\frac{7 π}{3} + 4 πn : \frac{7 π + 12 πn}{3}

\frac{7 π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{7 π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{7 π + 12 πn}{3}

= \frac{7 π + 12 πn}{3}

Simplify

- \frac{7 π}{3} + 4 πn : - \frac{7 π + 12 πn}{3}

- \frac{7 π}{3} + 4 πn

Join

\frac{7 π}{3} + 4 πn : \frac{7 π + 12 πn}{3}

\frac{7 π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{7 π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{7 π + 12 πn}{3}

= - \frac{7 π + 12 πn}{3}

x = \frac{7 π + 12 πn}{3}, x = - \frac{7 π + 12 πn}{3}

Solve

\frac{x ^{2}}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π + 12 πn}{3}, x = - \frac{11 π + 12 πn}{3}

\frac{x ^{2}}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{x ^{2}}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x ^{2}}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x ^{2}}{2} = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x ^{2}}{2} : x^{2}

\frac{2 x ^{2}}{2}

Divide the numbers:

\frac{2}{2} = 1

= x^{2}

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x^{2} = \frac{11 π}{3} + 4 πn

x^{2} = \frac{11 π}{3} + 4 πn

x^{2} = \frac{11 π}{3} + 4 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{11 π}{3} + 4 πn, x = - \frac{11 π}{3} + 4 πn

Simplify

\frac{11 π}{3} + 4 πn : \frac{11 π + 12 πn}{3}

\frac{11 π}{3} + 4 πn

Join

\frac{11 π}{3} + 4 πn : \frac{11 π + 12 πn}{3}

\frac{11 π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{11 π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{11 π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{11 π + 12 πn}{3}

= \frac{11 π + 12 πn}{3}

Simplify

- \frac{11 π}{3} + 4 πn : - \frac{11 π + 12 πn}{3}

- \frac{11 π}{3} + 4 πn

Join

\frac{11 π}{3} + 4 πn : \frac{11 π + 12 πn}{3}

\frac{11 π}{3} + 4 πn

Convert element to fraction:

4 πn = \frac{4 πn 3}{3}

= \frac{11 π}{3} + \frac{4 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{11 π + 4 πn \cdot 3}{3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{11 π + 12 πn}{3}

= - \frac{11 π + 12 πn}{3}

x = \frac{11 π + 12 πn}{3}, x = - \frac{11 π + 12 πn}{3}

x = \frac{7 π + 12 πn}{3}, x = - \frac{7 π + 12 πn}{3}, x = \frac{11 π + 12 πn}{3}, x = - \frac{11 π + 12 πn}{3}

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