What is the general solution for sin(2x+3/pi)= 1/2 ?

The general solution for sin(2x+3/pi)= 1/2 is x=pin+pi/(12)-3/(2pi),x=pin-3/(2pi)+(5pi)/(12)

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sin(2x+3/pi)= 1/2

Solution

sin (2 x + \frac{3}{π}) = \frac{1}{2}

Solution

x = πn + \frac{π}{12} - \frac{3}{2 π}, x = πn - \frac{3}{2 π} + \frac{5 π}{12}

Degrees

x = - 12.35671 \dots^{\circ} + 18 0^{\circ} n, x = 47.64328 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 x + \frac{3}{π}) = \frac{1}{2}

General solutions for

sin (2 x + \frac{3}{π}) = \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}

2 x + \frac{3}{π} = \frac{π}{6} + 2 πn, 2 x + \frac{3}{π} = \frac{5 π}{6} + 2 πn

2 x + \frac{3}{π} = \frac{π}{6} + 2 πn, 2 x + \frac{3}{π} = \frac{5 π}{6} + 2 πn

Solve

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

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

Move

\frac{3}{π}

to the right side

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

Subtract

\frac{3}{π}

from both sides

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

Simplify

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

Simplify

2 x + \frac{3}{π} - \frac{3}{π} : 2 x

2 x + \frac{3}{π} - \frac{3}{π}

Add similar elements:

\frac{3}{π} - \frac{3}{π} = 0

= 2 x

Simplify

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

\frac{π}{6} + 2 πn - \frac{3}{π}

Group like terms

= 2 πn + \frac{π}{6} - \frac{3}{π}

Could not simplify further

= 2 πn + \frac{π}{6} - \frac{3}{π}

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

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{2 πn}{2} + \frac{\frac{π}{6}}{2} - \frac{\frac{3}{π}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{\frac{3}{π}}{2} = \frac{3}{2 π}

\frac{\frac{3}{π}}{2}

Apply the fraction rule:

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

= \frac{3}{π 2}

= πn + \frac{π}{12} - \frac{3}{2 π}

x = πn + \frac{π}{12} - \frac{3}{2 π}

x = πn + \frac{π}{12} - \frac{3}{2 π}

x = πn + \frac{π}{12} - \frac{3}{2 π}

Solve

2 x + \frac{3}{π} = \frac{5 π}{6} + 2 πn : x = πn - \frac{3}{2 π} + \frac{5 π}{12}

2 x + \frac{3}{π} = \frac{5 π}{6} + 2 πn

Move

\frac{3}{π}

to the right side

2 x + \frac{3}{π} = \frac{5 π}{6} + 2 πn

Subtract

\frac{3}{π}

from both sides

2 x + \frac{3}{π} - \frac{3}{π} = \frac{5 π}{6} + 2 πn - \frac{3}{π}

Simplify

2 x + \frac{3}{π} - \frac{3}{π} = \frac{5 π}{6} + 2 πn - \frac{3}{π}

Simplify

2 x + \frac{3}{π} - \frac{3}{π} : 2 x

2 x + \frac{3}{π} - \frac{3}{π}

Add similar elements:

\frac{3}{π} - \frac{3}{π} = 0

= 2 x

Simplify

\frac{5 π}{6} + 2 πn - \frac{3}{π} : 2 πn - \frac{3}{π} + \frac{5 π}{6}

\frac{5 π}{6} + 2 πn - \frac{3}{π}

Group like terms

= 2 πn - \frac{3}{π} + \frac{5 π}{6}

Could not simplify further

= 2 πn - \frac{3}{π} + \frac{5 π}{6}

2 x = 2 πn - \frac{3}{π} + \frac{5 π}{6}

2 x = 2 πn - \frac{3}{π} + \frac{5 π}{6}

2 x = 2 πn - \frac{3}{π} + \frac{5 π}{6}

Divide both sides by

2

2 x = 2 πn - \frac{3}{π} + \frac{5 π}{6}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} - \frac{\frac{3}{π}}{2} + \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} - \frac{\frac{3}{π}}{2} + \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} - \frac{\frac{3}{π}}{2} + \frac{\frac{5 π}{6}}{2} : πn - \frac{3}{2 π} + \frac{5 π}{12}

\frac{2 πn}{2} - \frac{\frac{3}{π}}{2} + \frac{\frac{5 π}{6}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{3}{π}}{2} = \frac{3}{2 π}

\frac{\frac{3}{π}}{2}

Apply the fraction rule:

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

= \frac{3}{π 2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

= πn - \frac{3}{2 π} + \frac{5 π}{12}

x = πn - \frac{3}{2 π} + \frac{5 π}{12}

x = πn - \frac{3}{2 π} + \frac{5 π}{12}

x = πn - \frac{3}{2 π} + \frac{5 π}{12}

x = πn + \frac{π}{12} - \frac{3}{2 π}, x = πn - \frac{3}{2 π} + \frac{5 π}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+3/pi)= 1/2 ?
The general solution for sin(2x+3/pi)= 1/2 is x=pin+pi/(12)-3/(2pi),x=pin-3/(2pi)+(5pi)/(12)