What is the general solution for 3sin(2x-1)=1 ?

The general solution for 3sin(2x-1)=1 is x=pin+1/2+(0.33983…)/2 ,x=pin+pi/2+1/2-(0.33983…)/2

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3sin(2x-1)=1

Solution

3 sin (2 x - 1) = 1

Solution

x = πn + \frac{1}{2} + \frac{0.33983 \dots}{2}, x = πn + \frac{π}{2} + \frac{1}{2} - \frac{0.33983 \dots}{2}

Degrees

x = 38.38350 \dots^{\circ} + 18 0^{\circ} n, x = 108.91227 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sin (2 x - 1) = 1

Divide both sides by

3

3 sin (2 x - 1) = 1

Divide both sides by

3

\frac{3 sin ( 2 x - 1 )}{3} = \frac{1}{3}

Simplify

sin (2 x - 1) = \frac{1}{3}

sin (2 x - 1) = \frac{1}{3}

Apply trig inverse properties

sin (2 x - 1) = \frac{1}{3}

General solutions for

sin (2 x - 1) = \frac{1}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x - 1 = arcsin (\frac{1}{3}) + 2 πn, 2 x - 1 = π - arcsin (\frac{1}{3}) + 2 πn

2 x - 1 = arcsin (\frac{1}{3}) + 2 πn, 2 x - 1 = π - arcsin (\frac{1}{3}) + 2 πn

Solve

2 x - 1 = arcsin (\frac{1}{3}) + 2 πn : x = πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

2 x - 1 = arcsin (\frac{1}{3}) + 2 πn

Move

1

to the right side

2 x - 1 = arcsin (\frac{1}{3}) + 2 πn

Add

1

to both sides

2 x - 1 + 1 = arcsin (\frac{1}{3}) + 2 πn + 1

Simplify

2 x = arcsin (\frac{1}{3}) + 2 πn + 1

2 x = arcsin (\frac{1}{3}) + 2 πn + 1

Divide both sides by

2

2 x = arcsin (\frac{1}{3}) + 2 πn + 1

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2} : πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

\frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Group like terms

= \frac{1}{2} + \frac{2 πn}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{2} + πn + \frac{arcsin ( \frac{1}{3} )}{2}

Group like terms

= πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

Solve

2 x - 1 = π - arcsin (\frac{1}{3}) + 2 πn : x = πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

2 x - 1 = π - arcsin (\frac{1}{3}) + 2 πn

Move

1

to the right side

2 x - 1 = π - arcsin (\frac{1}{3}) + 2 πn

Add

1

to both sides

2 x - 1 + 1 = π - arcsin (\frac{1}{3}) + 2 πn + 1

Simplify

2 x = π - arcsin (\frac{1}{3}) + 2 πn + 1

2 x = π - arcsin (\frac{1}{3}) + 2 πn + 1

Divide both sides by

2

2 x = π - arcsin (\frac{1}{3}) + 2 πn + 1

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2} : πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

\frac{π}{2} - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Group like terms

= \frac{π}{2} + \frac{1}{2} + \frac{2 πn}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π}{2} + \frac{1}{2} + πn - \frac{arcsin ( \frac{1}{3} )}{2}

Group like terms

= πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

x = πn + \frac{1}{2} + \frac{arcsin ( \frac{1}{3} )}{2}, x = πn + \frac{π}{2} + \frac{1}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

Show solutions in decimal form

x = πn + \frac{1}{2} + \frac{0.33983 \dots}{2}, x = πn + \frac{π}{2} + \frac{1}{2} - \frac{0.33983 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(2x-1)=1 ?
The general solution for 3sin(2x-1)=1 is x=pin+1/2+(0.33983…)/2 ,x=pin+pi/2+1/2-(0.33983…)/2