What is the general solution for sin(3x-6)= 1/2 ?

The general solution for sin(3x-6)= 1/2 is x=(360n)/3+12,x=(360n)/3+52

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sin(3x-6)= 1/2

Solution

sin (3 x - 6^{\circ}) = \frac{1}{2}

Solution

x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}, x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

Radians

x = \frac{π}{15} + \frac{2 π}{3} n, x = \frac{13 π}{45} + \frac{2 π}{3} n

Solution steps

sin (3 x - 6^{\circ}) = \frac{1}{2}

General solutions for

sin (3 x - 6^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

3 x - 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n, 3 x - 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

3 x - 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n, 3 x - 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Solve

3 x - 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

3 x - 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n

Move

6^{\circ}

to the right side

3 x - 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n

Add

6^{\circ}

to both sides

3 x - 6^{\circ} + 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Simplify

3 x - 6^{\circ} + 6^{\circ} = 3 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Simplify

3 x - 6^{\circ} + 6^{\circ} : 3 x

3 x - 6^{\circ} + 6^{\circ}

Add similar elements:

- 6^{\circ} + 6^{\circ} = 0

= 3 x

Simplify

3 0^{\circ} + 36 0^{\circ} n + 6^{\circ} : 36 0^{\circ} n + 3 6^{\circ}

3 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Group like terms

= 36 0^{\circ} n + 3 0^{\circ} + 6^{\circ}

Least Common Multiplier of

6, 30 : 30

6, 30

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

30 : 2 \cdot 3 \cdot 5

30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 5

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

6

30

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

For

3 0^{\circ} :

multiply the denominator and numerator by

5

3 0^{\circ} = \frac{18 0 ^{\circ} 5}{6 \cdot 5} = 3 0^{\circ}

= 3 0^{\circ} + 6^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 5 + 18 0 ^{\circ}}{30}

Add similar elements:

90 0^{\circ} + 18 0^{\circ} = 108 0^{\circ}

= 3 6^{\circ}

Cancel the common factor:

6

= 36 0^{\circ} n + 3 6^{\circ}

3 x = 36 0^{\circ} n + 3 6^{\circ}

3 x = 36 0^{\circ} n + 3 6^{\circ}

3 x = 36 0^{\circ} n + 3 6^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 3 6^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{3 6 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{3 6 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{3 6 ^{\circ}}{3} : \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

\frac{36 0 ^{\circ} n}{3} + \frac{3 6 ^{\circ}}{3}

\frac{3 6 ^{\circ}}{3} = 1 2^{\circ}

\frac{3 6 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{5 \cdot 3}

Multiply the numbers:

5 \cdot 3 = 15

= 1 2^{\circ}

= \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}

Solve

3 x - 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

3 x - 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Move

6^{\circ}

to the right side

3 x - 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Add

6^{\circ}

to both sides

3 x - 6^{\circ} + 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Simplify

3 x - 6^{\circ} + 6^{\circ} = 15 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Simplify

3 x - 6^{\circ} + 6^{\circ} : 3 x

3 x - 6^{\circ} + 6^{\circ}

Add similar elements:

- 6^{\circ} + 6^{\circ} = 0

= 3 x

Simplify

15 0^{\circ} + 36 0^{\circ} n + 6^{\circ} : 36 0^{\circ} n + 15 6^{\circ}

15 0^{\circ} + 36 0^{\circ} n + 6^{\circ}

Group like terms

= 36 0^{\circ} n + 15 0^{\circ} + 6^{\circ}

Least Common Multiplier of

6, 30 : 30

6, 30

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

30 : 2 \cdot 3 \cdot 5

30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 5

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

6

30

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

For

15 0^{\circ} :

multiply the denominator and numerator by

5

15 0^{\circ} = \frac{90 0 ^{\circ} 5}{6 \cdot 5} = 15 0^{\circ}

= 15 0^{\circ} + 6^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{450 0 ^{\circ} + 18 0 ^{\circ}}{30}

Add similar elements:

450 0^{\circ} + 18 0^{\circ} = 468 0^{\circ}

= 15 6^{\circ}

Cancel the common factor:

2

= 36 0^{\circ} n + 15 6^{\circ}

3 x = 36 0^{\circ} n + 15 6^{\circ}

3 x = 36 0^{\circ} n + 15 6^{\circ}

3 x = 36 0^{\circ} n + 15 6^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 15 6^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{15 6 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{15 6 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{15 6 ^{\circ}}{3} : \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

\frac{36 0 ^{\circ} n}{3} + \frac{15 6 ^{\circ}}{3}

\frac{15 6 ^{\circ}}{3} = 5 2^{\circ}

\frac{15 6 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{234 0 ^{\circ}}{15 \cdot 3}

Multiply the numbers:

15 \cdot 3 = 45

= 5 2^{\circ}

= \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 2^{\circ}, x = \frac{36 0 ^{\circ} n}{3} + 5 2^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x-6)= 1/2 ?
The general solution for sin(3x-6)= 1/2 is x=(360n)/3+12,x=(360n)/3+52