What is the general solution for 0=3sin(x-1/2 pi)+2 ?

The general solution for 0=3sin(x-1/2 pi)+2 is x=-0.72972…+2pin+pi/2 ,x=pi+0.72972…+2pin+pi/2

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0=3sin(x-1/2 pi)+2

Solution

0 = 3 sin (x - \frac{1}{2} π) + 2

Solution

x = - 0.72972 \dots + 2 πn + \frac{π}{2}, x = π + 0.72972 \dots + 2 πn + \frac{π}{2}

Degrees

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0 = 3 sin (x - \frac{1}{2} π) + 2

Switch sides

3 sin (x - \frac{1}{2} π) + 2 = 0

Move

2

to the right side

3 sin (x - \frac{1}{2} π) + 2 = 0

Subtract

2

from both sides

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

Simplify

3 sin (x - \frac{1}{2} π) = - 2

3 sin (x - \frac{1}{2} π) = - 2

Divide both sides by

3

3 sin (x - \frac{1}{2} π) = - 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

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

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

Simplify

x - \frac{1}{2} π : x - \frac{π}{2}

x - \frac{1}{2} π

\frac{1}{2} π = \frac{π}{2}

\frac{1}{2} π

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

= x - \frac{π}{2}

Simplify

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

arcsin (- \frac{2}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

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

Solve

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

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

\frac{1}{2} π = \frac{π}{2}

\frac{1}{2} π

Multiply fractions:

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

= \frac{1 π}{2}

Multiply:

1 π = π

= \frac{π}{2}

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

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

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

Show solutions in decimal form

x = - 0.72972 \dots + 2 πn + \frac{π}{2}, x = π + 0.72972 \dots + 2 πn + \frac{π}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for 0=3sin(x-1/2 pi)+2 ?
The general solution for 0=3sin(x-1/2 pi)+2 is x=-0.72972…+2pin+pi/2 ,x=pi+0.72972…+2pin+pi/2