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

The general solution for 3sin(pi/2 x)=2 is x=(2*0.72972…)/pi+4n,x=2-(2*0.72972…)/pi+4n

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3sin(pi/2 x)=2

Solution

3 sin (\frac{π}{2} x) = 2

Solution

x = \frac{2 \cdot 0.72972 \dots}{π} + 4 n, x = 2 - \frac{2 \cdot 0.72972 \dots}{π} + 4 n

Degrees

x = 26.61727 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 87.97428 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

3 sin (\frac{π}{2} x) = 2

Divide both sides by

3

3 sin (\frac{π}{2} x) = 2

Divide both sides by

3

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

Simplify

sin (\frac{π}{2} x) = \frac{2}{3}

sin (\frac{π}{2} x) = \frac{2}{3}

Apply trig inverse properties

sin (\frac{π}{2} x) = \frac{2}{3}

General solutions for

sin (\frac{π}{2} x) = \frac{2}{3}

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

Solve

\frac{π}{2} x = arcsin (\frac{2}{3}) + 2 πn : x = \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{π}{2} x : π x

2 \cdot \frac{π}{2} x

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

2 arcsin (\frac{2}{3}) + 2 \cdot 2 πn : 2 arcsin (\frac{2}{3}) + 4 πn

2 arcsin (\frac{2}{3}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 arcsin (\frac{2}{3}) + 4 πn

Divide both sides by

π

π x = 2 arcsin (\frac{2}{3}) + 4 πn

Divide both sides by

π

\frac{π x}{π} = \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π}

Simplify

x = \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

x = \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

Solve

\frac{π}{2} x = π - arcsin (\frac{2}{3}) + 2 πn : x = 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{π}{2} x : π x

2 \cdot \frac{π}{2} x

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

2 π - 2 arcsin (\frac{2}{3}) + 2 \cdot 2 πn : 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

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

Multiply the numbers:

2 \cdot 2 = 4

= 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

π x = 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

Divide both sides by

π

π x = 2 π - 2 arcsin (\frac{2}{3}) + 4 πn

Divide both sides by

π

\frac{π x}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π} : 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

\frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

x = 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

x = 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

x = 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

x = \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n, x = 2 - \frac{2 arcsin ( \frac{2}{3} )}{π} + 4 n

Show solutions in decimal form

x = \frac{2 \cdot 0.72972 \dots}{π} + 4 n, x = 2 - \frac{2 \cdot 0.72972 \dots}{π} + 4 n

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(pi/2 x)=2 ?
The general solution for 3sin(pi/2 x)=2 is x=(2*0.72972…)/pi+4n,x=2-(2*0.72972…)/pi+4n