What is the general solution for sin(2x-50)=-1/2 ?

The general solution for sin(2x-50)=-1/2 is x=180n+130,x=180n+190

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sin(2x-50)=-1/2

Solution

sin (2 x - 5 0^{\circ}) = - \frac{1}{2}

Solution

x = 18 0^{\circ} n + 13 0^{\circ}, x = 18 0^{\circ} n + 19 0^{\circ}

Radians

x = \frac{13 π}{18} + πn, x = \frac{19 π}{18} + πn

Solution steps

sin (2 x - 5 0^{\circ}) = - \frac{1}{2}

General solutions for

sin (2 x - 5 0^{\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}

2 x - 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n, 2 x - 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n

2 x - 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n, 2 x - 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n

Solve

2 x - 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 13 0^{\circ}

2 x - 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n

Move

5 0^{\circ}

to the right side

2 x - 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n

Add

5 0^{\circ}

to both sides

2 x - 5 0^{\circ} + 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Simplify

2 x - 5 0^{\circ} + 5 0^{\circ} = 21 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Simplify

2 x - 5 0^{\circ} + 5 0^{\circ} : 2 x

2 x - 5 0^{\circ} + 5 0^{\circ}

Add similar elements:

- 5 0^{\circ} + 5 0^{\circ} = 0

= 2 x

Simplify

21 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ} : 36 0^{\circ} n + 26 0^{\circ}

21 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Group like terms

= 36 0^{\circ} n + 21 0^{\circ} + 5 0^{\circ}

Least Common Multiplier of

6, 18 : 18

6, 18

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

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

6

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

21 0^{\circ} :

multiply the denominator and numerator by

3

21 0^{\circ} = \frac{126 0 ^{\circ} 3}{6 \cdot 3} = 21 0^{\circ}

= 21 0^{\circ} + 5 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{378 0 ^{\circ} + 90 0 ^{\circ}}{18}

Add similar elements:

378 0^{\circ} + 90 0^{\circ} = 468 0^{\circ}

= 26 0^{\circ}

Cancel the common factor:

2

= 36 0^{\circ} n + 26 0^{\circ}

2 x = 36 0^{\circ} n + 26 0^{\circ}

2 x = 36 0^{\circ} n + 26 0^{\circ}

2 x = 36 0^{\circ} n + 26 0^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} n + 26 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{26 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{26 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{26 0 ^{\circ}}{2} : 18 0^{\circ} n + 13 0^{\circ}

\frac{36 0 ^{\circ} n}{2} + \frac{26 0 ^{\circ}}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{26 0 ^{\circ}}{2} = 13 0^{\circ}

\frac{26 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{234 0 ^{\circ}}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= 13 0^{\circ}

= 18 0^{\circ} n + 13 0^{\circ}

x = 18 0^{\circ} n + 13 0^{\circ}

x = 18 0^{\circ} n + 13 0^{\circ}

x = 18 0^{\circ} n + 13 0^{\circ}

Solve

2 x - 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 19 0^{\circ}

2 x - 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n

Move

5 0^{\circ}

to the right side

2 x - 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n

Add

5 0^{\circ}

to both sides

2 x - 5 0^{\circ} + 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Simplify

2 x - 5 0^{\circ} + 5 0^{\circ} = 33 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Simplify

2 x - 5 0^{\circ} + 5 0^{\circ} : 2 x

2 x - 5 0^{\circ} + 5 0^{\circ}

Add similar elements:

- 5 0^{\circ} + 5 0^{\circ} = 0

= 2 x

Simplify

33 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ} : 36 0^{\circ} n + 38 0^{\circ}

33 0^{\circ} + 36 0^{\circ} n + 5 0^{\circ}

Group like terms

= 36 0^{\circ} n + 33 0^{\circ} + 5 0^{\circ}

Least Common Multiplier of

6, 18 : 18

6, 18

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

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

6

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

33 0^{\circ} :

multiply the denominator and numerator by

3

33 0^{\circ} = \frac{198 0 ^{\circ} 3}{6 \cdot 3} = 33 0^{\circ}

= 33 0^{\circ} + 5 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{594 0 ^{\circ} + 90 0 ^{\circ}}{18}

Add similar elements:

594 0^{\circ} + 90 0^{\circ} = 684 0^{\circ}

= 38 0^{\circ}

Cancel the common factor:

2

= 36 0^{\circ} n + 38 0^{\circ}

2 x = 36 0^{\circ} n + 38 0^{\circ}

2 x = 36 0^{\circ} n + 38 0^{\circ}

2 x = 36 0^{\circ} n + 38 0^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} n + 38 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{38 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{38 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{38 0 ^{\circ}}{2} : 18 0^{\circ} n + 19 0^{\circ}

\frac{36 0 ^{\circ} n}{2} + \frac{38 0 ^{\circ}}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{38 0 ^{\circ}}{2} = 19 0^{\circ}

\frac{38 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{342 0 ^{\circ}}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= 19 0^{\circ}

= 18 0^{\circ} n + 19 0^{\circ}

x = 18 0^{\circ} n + 19 0^{\circ}

x = 18 0^{\circ} n + 19 0^{\circ}

x = 18 0^{\circ} n + 19 0^{\circ}

x = 18 0^{\circ} n + 13 0^{\circ}, x = 18 0^{\circ} n + 19 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x-50)=-1/2 ?
The general solution for sin(2x-50)=-1/2 is x=180n+130,x=180n+190