What is the general solution for sin(x+pi/9)=(sqrt(3))/2 ?

The general solution for sin(x+pi/9)=(sqrt(3))/2 is x=2pin+(2pi)/9 ,x=2pin+(5pi)/9

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sin(x+pi/9)=(sqrt(3))/2

Solution

sin (x + \frac{π}{9}) = \frac{3}{2}

Solution

x = 2 πn + \frac{2 π}{9}, x = 2 πn + \frac{5 π}{9}

Degrees

x = 4 0^{\circ} + 36 0^{\circ} n, x = 10 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x + \frac{π}{9}) = \frac{3}{2}

General solutions for

sin (x + \frac{π}{9}) = \frac{3}{2}

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}

x + \frac{π}{9} = \frac{π}{3} + 2 πn, x + \frac{π}{9} = \frac{2 π}{3} + 2 πn

x + \frac{π}{9} = \frac{π}{3} + 2 πn, x + \frac{π}{9} = \frac{2 π}{3} + 2 πn

Solve

x + \frac{π}{9} = \frac{π}{3} + 2 πn : x = 2 πn + \frac{2 π}{9}

x + \frac{π}{9} = \frac{π}{3} + 2 πn

Move

\frac{π}{9}

to the right side

x + \frac{π}{9} = \frac{π}{3} + 2 πn

Subtract

\frac{π}{9}

from both sides

x + \frac{π}{9} - \frac{π}{9} = \frac{π}{3} + 2 πn - \frac{π}{9}

Simplify

x + \frac{π}{9} - \frac{π}{9} = \frac{π}{3} + 2 πn - \frac{π}{9}

Simplify

x + \frac{π}{9} - \frac{π}{9} : x

x + \frac{π}{9} - \frac{π}{9}

Add similar elements:

\frac{π}{9} - \frac{π}{9} = 0

= x

Simplify

\frac{π}{3} + 2 πn - \frac{π}{9} : 2 πn + \frac{2 π}{9}

\frac{π}{3} + 2 πn - \frac{π}{9}

Group like terms

= 2 πn + \frac{π}{3} - \frac{π}{9}

Least Common Multiplier of

3, 9 : 9

3, 9

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

3

9

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

9

For

\frac{π}{3} :

multiply the denominator and numerator by

3

\frac{π}{3} = \frac{π 3}{3 \cdot 3} = \frac{π 3}{9}

= \frac{π 3}{9} - \frac{π}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{9}

Add similar elements:

3 π - π = 2 π

= 2 πn + \frac{2 π}{9}

x = 2 πn + \frac{2 π}{9}

x = 2 πn + \frac{2 π}{9}

x = 2 πn + \frac{2 π}{9}

Solve

x + \frac{π}{9} = \frac{2 π}{3} + 2 πn : x = 2 πn + \frac{5 π}{9}

x + \frac{π}{9} = \frac{2 π}{3} + 2 πn

Move

\frac{π}{9}

to the right side

x + \frac{π}{9} = \frac{2 π}{3} + 2 πn

Subtract

\frac{π}{9}

from both sides

x + \frac{π}{9} - \frac{π}{9} = \frac{2 π}{3} + 2 πn - \frac{π}{9}

Simplify

x + \frac{π}{9} - \frac{π}{9} = \frac{2 π}{3} + 2 πn - \frac{π}{9}

Simplify

x + \frac{π}{9} - \frac{π}{9} : x

x + \frac{π}{9} - \frac{π}{9}

Add similar elements:

\frac{π}{9} - \frac{π}{9} = 0

= x

Simplify

\frac{2 π}{3} + 2 πn - \frac{π}{9} : 2 πn + \frac{5 π}{9}

\frac{2 π}{3} + 2 πn - \frac{π}{9}

Group like terms

= 2 πn - \frac{π}{9} + \frac{2 π}{3}

Least Common Multiplier of

9, 3 : 9

9, 3

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

9

3

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

9

For

\frac{2 π}{3} :

multiply the denominator and numerator by

3

\frac{2 π}{3} = \frac{2 π 3}{3 \cdot 3} = \frac{6 π}{9}

= - \frac{π}{9} + \frac{6 π}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + 6 π}{9}

Add similar elements:

- π + 6 π = 5 π

= 2 πn + \frac{5 π}{9}

x = 2 πn + \frac{5 π}{9}

x = 2 πn + \frac{5 π}{9}

x = 2 πn + \frac{5 π}{9}

x = 2 πn + \frac{2 π}{9}, x = 2 πn + \frac{5 π}{9}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+pi/9)=(sqrt(3))/2 ?
The general solution for sin(x+pi/9)=(sqrt(3))/2 is x=2pin+(2pi)/9 ,x=2pin+(5pi)/9