What is the general solution for sin(2x+3)=0.5 ?

The general solution for sin(2x+3)=0.5 is x=pin-3/2+pi/(12),x=pin-3/2+(5pi)/(12)

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sin(2x+3)=0.5

Solution

sin (2 x + 3) = 0.5

Solution

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

Degrees

x = - 70.94366 \dots^{\circ} + 18 0^{\circ} n, x = - 10.94366 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 x + 3) = 0.5

General solutions for

sin (2 x + 3) = 0.5

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

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

Solve

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

2 x + 3 = \frac{π}{6} + 2 πn

Move

3

to the right side

2 x + 3 = \frac{π}{6} + 2 πn

Subtract

3

from both sides

2 x + 3 - 3 = \frac{π}{6} + 2 πn - 3

Simplify

2 x = \frac{π}{6} + 2 πn - 3

2 x = \frac{π}{6} + 2 πn - 3

Divide both sides by

2

2 x = \frac{π}{6} + 2 πn - 3

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

Group like terms

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

Group like terms

= πn - \frac{3}{2} + \frac{π}{12}

x = πn - \frac{3}{2} + \frac{π}{12}

x = πn - \frac{3}{2} + \frac{π}{12}

x = πn - \frac{3}{2} + \frac{π}{12}

Solve

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

2 x + 3 = \frac{5 π}{6} + 2 πn

Move

3

to the right side

2 x + 3 = \frac{5 π}{6} + 2 πn

Subtract

3

from both sides

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

Simplify

2 x = \frac{5 π}{6} + 2 πn - 3

2 x = \frac{5 π}{6} + 2 πn - 3

Divide both sides by

2

2 x = \frac{5 π}{6} + 2 πn - 3

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

Group like terms

= - \frac{3}{2} + \frac{2 πn}{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{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}

= - \frac{3}{2} + πn + \frac{5 π}{12}

Group like terms

= πn - \frac{3}{2} + \frac{5 π}{12}

x = πn - \frac{3}{2} + \frac{5 π}{12}

x = πn - \frac{3}{2} + \frac{5 π}{12}

x = πn - \frac{3}{2} + \frac{5 π}{12}

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+3)=0.5 ?
The general solution for sin(2x+3)=0.5 is x=pin-3/2+pi/(12),x=pin-3/2+(5pi)/(12)