What is the general solution for sin(3θ+pi/(18))=1,0<= θ<2pi ?

Q: What is the general solution for sin(3θ+pi/(18))=1,0<= θ<2pi ?

The general solution for sin(3θ+pi/(18))=1,0<= θ<2pi is θ=(4pi)/(27),θ=(22pi)/(27),θ=(40pi)/(27)

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sin(3θ+pi/(18))=1,0<= θ<2pi

Solution

sin (3 θ + \frac{π}{18}) = 1, 0 \leq θ < 2 π

Solution

θ = \frac{4 π}{27}, θ = \frac{22 π}{27}, θ = \frac{40 π}{27}

Degrees

θ = 26.66666 \dots^{\circ}, θ = 146.66666 \dots^{\circ}, θ = 266.66666 \dots^{\circ}

Solution steps

sin (3 θ + \frac{π}{18}) = 1, 0 \leq θ < 2 π

General solutions for

sin (3 θ + \frac{π}{18}) = 1

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}

3 θ + \frac{π}{18} = \frac{π}{2} + 2 πn

3 θ + \frac{π}{18} = \frac{π}{2} + 2 πn

Solve

3 θ + \frac{π}{18} = \frac{π}{2} + 2 πn : θ = \frac{2 πn}{3} + \frac{4 π}{27}

3 θ + \frac{π}{18} = \frac{π}{2} + 2 πn

Move

\frac{π}{18}

to the right side

3 θ + \frac{π}{18} = \frac{π}{2} + 2 πn

Subtract

\frac{π}{18}

from both sides

3 θ + \frac{π}{18} - \frac{π}{18} = \frac{π}{2} + 2 πn - \frac{π}{18}

Simplify

3 θ + \frac{π}{18} - \frac{π}{18} = \frac{π}{2} + 2 πn - \frac{π}{18}

Simplify

3 θ + \frac{π}{18} - \frac{π}{18} : 3 θ

3 θ + \frac{π}{18} - \frac{π}{18}

Add similar elements:

\frac{π}{18} - \frac{π}{18} = 0

= 3 θ

Simplify

\frac{π}{2} + 2 πn - \frac{π}{18} : 2 πn + \frac{4 π}{9}

\frac{π}{2} + 2 πn - \frac{π}{18}

Group like terms

= 2 πn + \frac{π}{2} - \frac{π}{18}

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

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

2

18

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

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

18

For

\frac{π}{2} :

multiply the denominator and numerator by

9

\frac{π}{2} = \frac{π 9}{2 \cdot 9} = \frac{π 9}{18}

= \frac{π 9}{18} - \frac{π}{18}

Since the denominators are equal, combine the fractions:

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

= \frac{π 9 - π}{18}

Add similar elements:

9 π - π = 8 π

= \frac{8 π}{18}

Cancel the common factor:

2

= 2 πn + \frac{4 π}{9}

3 θ = 2 πn + \frac{4 π}{9}

3 θ = 2 πn + \frac{4 π}{9}

3 θ = 2 πn + \frac{4 π}{9}

Divide both sides by

3

3 θ = 2 πn + \frac{4 π}{9}

Divide both sides by

3

\frac{3 θ}{3} = \frac{2 πn}{3} + \frac{\frac{4 π}{9}}{3}

Simplify

\frac{3 θ}{3} = \frac{2 πn}{3} + \frac{\frac{4 π}{9}}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{2 πn}{3} + \frac{\frac{4 π}{9}}{3} : \frac{2 πn}{3} + \frac{4 π}{27}

\frac{2 πn}{3} + \frac{\frac{4 π}{9}}{3}

\frac{\frac{4 π}{9}}{3} = \frac{4 π}{27}

\frac{\frac{4 π}{9}}{3}

Apply the fraction rule:

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

= \frac{4 π}{9 \cdot 3}

Multiply the numbers:

9 \cdot 3 = 27

= \frac{4 π}{27}

= \frac{2 πn}{3} + \frac{4 π}{27}

θ = \frac{2 πn}{3} + \frac{4 π}{27}

θ = \frac{2 πn}{3} + \frac{4 π}{27}

θ = \frac{2 πn}{3} + \frac{4 π}{27}

θ = \frac{2 πn}{3} + \frac{4 π}{27}

Solutions for the range

0 \leq θ < 2 π

θ = \frac{4 π}{27}, θ = \frac{22 π}{27}, θ = \frac{40 π}{27}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3θ+pi/(18))=1,0<= θ<2pi ?
The general solution for sin(3θ+pi/(18))=1,0<= θ<2pi is θ=(4pi)/(27),θ=(22pi)/(27),θ=(40pi)/(27)