What is the general solution for sqrt(2)sin(x/2+pi/6)=1,x<4pi,0 ?

Q: What is the general solution for sqrt(2)sin(x/2+pi/6)=1,x<4pi,0 ?

The general solution for sqrt(2)sin(x/2+pi/6)=1,x<4pi,0 is

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sqrt(2)sin(x/2+pi/6)=1,x<4pi,0

Solution

2 sin (\frac{x}{2} + \frac{π}{6}) = 1, x < 4 π, 0

Solution

x = \frac{π}{6}, x = \frac{7 π}{6}, x = - \frac{23 π}{6}, x = - \frac{17 π}{6}, x = - \frac{47 π}{6}, x = - \frac{41 π}{6}, x = - \frac{71 π}{6}, x = - \frac{65 π}{6}, x = - \frac{95 π}{6}, x = - \frac{89 π}{6}

Degrees

x = 3 0^{\circ}, x = 21 0^{\circ}, x = - 69 0^{\circ}, x = - 51 0^{\circ}, x = - 141 0^{\circ}, x = - 123 0^{\circ}, x = - 213 0^{\circ}, x = - 195 0^{\circ}, x = - 285 0^{\circ}, x = - 267 0^{\circ}

Solution steps

2 sin (\frac{x}{2} + \frac{π}{6}) = 1, x < 4 π, 0

Divide both sides by

2

2 sin (\frac{x}{2} + \frac{π}{6}) = 1

Divide both sides by

2

\frac{2 sin ( \frac{x}{2} + \frac{π}{6} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( \frac{x}{2} + \frac{π}{6} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( \frac{x}{2} + \frac{π}{6} )}{2} : sin (\frac{x}{2} + \frac{π}{6})

\frac{2 sin ( \frac{x}{2} + \frac{π}{6} )}{2}

Cancel the common factor:

2

= sin (\frac{x}{2} + \frac{π}{6})

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (\frac{x}{2} + \frac{π}{6}) = \frac{2}{2}

sin (\frac{x}{2} + \frac{π}{6}) = \frac{2}{2}

sin (\frac{x}{2} + \frac{π}{6}) = \frac{2}{2}

General solutions for

sin (\frac{x}{2} + \frac{π}{6}) = \frac{2}{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} + \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{x}{2} + \frac{π}{6} = \frac{3 π}{4} + 2 πn

\frac{x}{2} + \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{x}{2} + \frac{π}{6} = \frac{3 π}{4} + 2 πn

Solve

\frac{x}{2} + \frac{π}{6} = \frac{π}{4} + 2 πn : x = 4 πn + \frac{π}{6}

\frac{x}{2} + \frac{π}{6} = \frac{π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{x}{2} + \frac{π}{6} = \frac{π}{4} + 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} = \frac{π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} = \frac{π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} : \frac{x}{2}

\frac{x}{2} + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= \frac{x}{2}

Simplify

\frac{π}{4} + 2 πn - \frac{π}{6} : 2 πn + \frac{π}{12}

\frac{π}{4} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{4} - \frac{π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= \frac{π 3}{12} - \frac{π 2}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π 2}{12}

Add similar elements:

3 π - 2 π = π

= 2 πn + \frac{π}{12}

\frac{x}{2} = 2 πn + \frac{π}{12}

\frac{x}{2} = 2 πn + \frac{π}{12}

\frac{x}{2} = 2 πn + \frac{π}{12}

Multiply both sides by

2

\frac{x}{2} = 2 πn + \frac{π}{12}

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{π}{12}

Simplify

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{π}{12}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 πn + 2 \cdot \frac{π}{12} : 4 πn + \frac{π}{6}

2 \cdot 2 πn + 2 \cdot \frac{π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{12} = \frac{π}{6}

2 \cdot \frac{π}{12}

Multiply fractions:

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

= \frac{π 2}{12}

Cancel the common factor:

2

= \frac{π}{6}

= 4 πn + \frac{π}{6}

x = 4 πn + \frac{π}{6}

x = 4 πn + \frac{π}{6}

x = 4 πn + \frac{π}{6}

Solve

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

\frac{x}{2} + \frac{π}{6} = \frac{3 π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{x}{2} + \frac{π}{6} = \frac{3 π}{4} + 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} = \frac{3 π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} = \frac{3 π}{4} + 2 πn - \frac{π}{6}

Simplify

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} : \frac{x}{2}

\frac{x}{2} + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= \frac{x}{2}

Simplify

\frac{3 π}{4} + 2 πn - \frac{π}{6} : 2 πn + \frac{7 π}{12}

\frac{3 π}{4} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn - \frac{π}{6} + \frac{3 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= - \frac{π 2}{12} + \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + 9 π}{12}

Add similar elements:

- 2 π + 9 π = 7 π

= 2 πn + \frac{7 π}{12}

\frac{x}{2} = 2 πn + \frac{7 π}{12}

\frac{x}{2} = 2 πn + \frac{7 π}{12}

\frac{x}{2} = 2 πn + \frac{7 π}{12}

Multiply both sides by

2

\frac{x}{2} = 2 πn + \frac{7 π}{12}

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{7 π}{12}

Simplify

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{7 π}{12}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 πn + 2 \cdot \frac{7 π}{12} : 4 πn + \frac{7 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{7 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{7 π}{12} = \frac{7 π}{6}

2 \cdot \frac{7 π}{12}

Multiply fractions:

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

= \frac{7 π 2}{12}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{12}

Cancel the common factor:

2

= \frac{7 π}{6}

= 4 πn + \frac{7 π}{6}

x = 4 πn + \frac{7 π}{6}

x = 4 πn + \frac{7 π}{6}

x = 4 πn + \frac{7 π}{6}

x = 4 πn + \frac{π}{6}, x = 4 πn + \frac{7 π}{6}

Solutions for the range

x < 4 π

x = \frac{π}{6}, x = \frac{7 π}{6}, x = - \frac{23 π}{6}, x = - \frac{17 π}{6}, x = - \frac{47 π}{6}, x = - \frac{41 π}{6}, x = - \frac{71 π}{6}, x = - \frac{65 π}{6}, x = - \frac{95 π}{6}, x = - \frac{89 π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin(x/2+pi/6)=1,x<4pi,0 ?
The general solution for sqrt(2)sin(x/2+pi/6)=1,x<4pi,0 is