What is the general solution for sin(5x-10)=(sqrt(2))/2 ?

The general solution for sin(5x-10)=(sqrt(2))/2 is x=(360n)/5+11,x=(360n)/5+29

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Popular Trigonometry >

sin(5x-10)=(sqrt(2))/2

Solution

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

Solution

x = \frac{36 0 ^{\circ} n}{5} + 1 1^{\circ}, x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

Radians

x = \frac{11 π}{180} + \frac{2 π}{5} n, x = \frac{29 π}{180} + \frac{2 π}{5} n

Solution steps

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

General solutions for

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

5 x - 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, 5 x - 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

5 x - 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, 5 x - 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Solve

5 x - 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{5} + 1 1^{\circ}

5 x - 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

5 x - 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Add

1 0^{\circ}

to both sides

5 x - 1 0^{\circ} + 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

5 x - 1 0^{\circ} + 1 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

5 x - 1 0^{\circ} + 1 0^{\circ} : 5 x

5 x - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 5 x

Simplify

4 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ} : 36 0^{\circ} n + 5 5^{\circ}

4 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Group like terms

= 36 0^{\circ} n + 4 5^{\circ} + 1 0^{\circ}

Least Common Multiplier of

4, 18 : 36

4, 18

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 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

4

18

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

4 5^{\circ} :

multiply the denominator and numerator by

9

4 5^{\circ} = \frac{18 0 ^{\circ} 9}{4 \cdot 9} = 4 5^{\circ}

For

1 0^{\circ} :

multiply the denominator and numerator by

2

1 0^{\circ} = \frac{18 0 ^{\circ} 2}{18 \cdot 2} = 1 0^{\circ}

= 4 5^{\circ} + 1 0^{\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} 9 + 18 0 ^{\circ} 2}{36}

Add similar elements:

162 0^{\circ} + 36 0^{\circ} = 198 0^{\circ}

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

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

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

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

Divide both sides by

5

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

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{5} + \frac{5 5 ^{\circ}}{5} : \frac{36 0 ^{\circ} n}{5} + 1 1^{\circ}

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

\frac{5 5 ^{\circ}}{5} = 1 1^{\circ}

\frac{5 5 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{198 0 ^{\circ}}{36 \cdot 5}

Multiply the numbers:

36 \cdot 5 = 180

= 1 1^{\circ}

= \frac{36 0 ^{\circ} n}{5} + 1 1^{\circ}

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

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

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

Solve

5 x - 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

5 x - 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

5 x - 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Add

1 0^{\circ}

to both sides

5 x - 1 0^{\circ} + 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

5 x - 1 0^{\circ} + 1 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Simplify

5 x - 1 0^{\circ} + 1 0^{\circ} : 5 x

5 x - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 5 x

Simplify

13 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ} : 36 0^{\circ} n + 14 5^{\circ}

13 5^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

Group like terms

= 36 0^{\circ} n + 13 5^{\circ} + 1 0^{\circ}

Least Common Multiplier of

4, 18 : 36

4, 18

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 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

4

18

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

13 5^{\circ} :

multiply the denominator and numerator by

9

13 5^{\circ} = \frac{54 0 ^{\circ} 9}{4 \cdot 9} = 13 5^{\circ}

For

1 0^{\circ} :

multiply the denominator and numerator by

2

1 0^{\circ} = \frac{18 0 ^{\circ} 2}{18 \cdot 2} = 1 0^{\circ}

= 13 5^{\circ} + 1 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{486 0 ^{\circ} + 18 0 ^{\circ} 2}{36}

Add similar elements:

486 0^{\circ} + 36 0^{\circ} = 522 0^{\circ}

= 36 0^{\circ} n + 14 5^{\circ}

5 x = 36 0^{\circ} n + 14 5^{\circ}

5 x = 36 0^{\circ} n + 14 5^{\circ}

5 x = 36 0^{\circ} n + 14 5^{\circ}

Divide both sides by

5

5 x = 36 0^{\circ} n + 14 5^{\circ}

Divide both sides by

5

\frac{5 x}{5} = \frac{36 0 ^{\circ} n}{5} + \frac{14 5 ^{\circ}}{5}

Simplify

\frac{5 x}{5} = \frac{36 0 ^{\circ} n}{5} + \frac{14 5 ^{\circ}}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{5} + \frac{14 5 ^{\circ}}{5} : \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

\frac{36 0 ^{\circ} n}{5} + \frac{14 5 ^{\circ}}{5}

\frac{14 5 ^{\circ}}{5} = 2 9^{\circ}

\frac{14 5 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{522 0 ^{\circ}}{36 \cdot 5}

Multiply the numbers:

36 \cdot 5 = 180

= 2 9^{\circ}

= \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

x = \frac{36 0 ^{\circ} n}{5} + 1 1^{\circ}, x = \frac{36 0 ^{\circ} n}{5} + 2 9^{\circ}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x-10)=(sqrt(2))/2 ?
The general solution for sin(5x-10)=(sqrt(2))/2 is x=(360n)/5+11,x=(360n)/5+29