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

The general solution for 3sin(pi/3 x)=2 is x=(3*0.72972…)/pi+6n,x=3-(3*0.72972…)/pi+6n

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3sin(pi/3 x)=2

Solution

3 sin (\frac{π}{3} x) = 2

Solution

x = \frac{3 \cdot 0.72972 \dots}{π} + 6 n, x = 3 - \frac{3 \cdot 0.72972 \dots}{π} + 6 n

Degrees

x = 39.92590 \dots^{\circ} + 343.77467 \dots^{\circ} n, x = 131.96142 \dots^{\circ} + 343.77467 \dots^{\circ} n

Solution steps

3 sin (\frac{π}{3} x) = 2

Divide both sides by

3

3 sin (\frac{π}{3} x) = 2

Divide both sides by

3

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

Simplify

sin (\frac{π}{3} x) = \frac{2}{3}

sin (\frac{π}{3} x) = \frac{2}{3}

Apply trig inverse properties

sin (\frac{π}{3} x) = \frac{2}{3}

General solutions for

sin (\frac{π}{3} x) = \frac{2}{3}

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

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

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

Solve

\frac{π}{3} x = arcsin (\frac{2}{3}) + 2 πn : x = \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{π}{3} x : π x

3 \cdot \frac{π}{3} x

Multiply fractions:

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

= \frac{3 π}{3} x

Cancel the common factor:

3

= x π

Simplify

3 arcsin (\frac{2}{3}) + 3 \cdot 2 πn : 3 arcsin (\frac{2}{3}) + 6 πn

3 arcsin (\frac{2}{3}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arcsin (\frac{2}{3}) + 6 πn

π x = 3 arcsin (\frac{2}{3}) + 6 πn

π x = 3 arcsin (\frac{2}{3}) + 6 πn

π x = 3 arcsin (\frac{2}{3}) + 6 πn

Divide both sides by

π

π x = 3 arcsin (\frac{2}{3}) + 6 πn

Divide both sides by

π

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

Simplify

x = \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

x = \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

Solve

\frac{π}{3} x = π - arcsin (\frac{2}{3}) + 2 πn : x = 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{π}{3} x : π x

3 \cdot \frac{π}{3} x

Multiply fractions:

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

= \frac{3 π}{3} x

Cancel the common factor:

3

= x π

Simplify

3 π - 3 arcsin (\frac{2}{3}) + 3 \cdot 2 πn : 3 π - 3 arcsin (\frac{2}{3}) + 6 πn

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

Multiply the numbers:

3 \cdot 2 = 6

= 3 π - 3 arcsin (\frac{2}{3}) + 6 πn

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

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

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

Divide both sides by

π

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

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{3 π}{π} - \frac{3 arcsin ( \frac{2}{3} )}{π} + \frac{6 πn}{π} : 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

\frac{3 π}{π} - \frac{3 arcsin ( \frac{2}{3} )}{π} + \frac{6 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + \frac{6 πn}{π}

Cancel

\frac{6 πn}{π} : 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

x = 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

x = 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

x = 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

x = \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n, x = 3 - \frac{3 arcsin ( \frac{2}{3} )}{π} + 6 n

Show solutions in decimal form

x = \frac{3 \cdot 0.72972 \dots}{π} + 6 n, x = 3 - \frac{3 \cdot 0.72972 \dots}{π} + 6 n

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(pi/3 x)=2 ?
The general solution for 3sin(pi/3 x)=2 is x=(3*0.72972…)/pi+6n,x=3-(3*0.72972…)/pi+6n