What is the general solution for sin(x-20)= 1/(sqrt(2)) ?

The general solution for sin(x-20)= 1/(sqrt(2)) is x=360n+65,x=360n+155

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sin(x-20)= 1/(sqrt(2))

Solution

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

Solution

x = 36 0^{\circ} n + 6 5^{\circ}, x = 36 0^{\circ} n + 15 5^{\circ}

Radians

x = \frac{13 π}{36} + 2 πn, x = \frac{31 π}{36} + 2 πn

Solution steps

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

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

General solutions for

sin (x - 2 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}

x - 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

x - 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Solve

x - 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 6 5^{\circ}

x - 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x - 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Add

2 0^{\circ}

to both sides

x - 2 0^{\circ} + 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} = 4 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

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

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

Add similar elements:

- 2 0^{\circ} + 2 0^{\circ} = 0

= x

Simplify

4 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 6 5^{\circ}

4 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 4 5^{\circ} + 2 0^{\circ}

Least Common Multiplier of

4, 9 : 36

4, 9

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

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

4

9

= 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

2 0^{\circ} :

multiply the denominator and numerator by

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

= 4 5^{\circ} + 2 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} 4}{36}

Add similar elements:

162 0^{\circ} + 72 0^{\circ} = 234 0^{\circ}

= 36 0^{\circ} n + 6 5^{\circ}

x = 36 0^{\circ} n + 6 5^{\circ}

x = 36 0^{\circ} n + 6 5^{\circ}

x = 36 0^{\circ} n + 6 5^{\circ}

Solve

x - 2 0^{\circ} = 13 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 15 5^{\circ}

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

Move

2 0^{\circ}

to the right side

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

Add

2 0^{\circ}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

- 2 0^{\circ} + 2 0^{\circ} = 0

= x

Simplify

13 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 15 5^{\circ}

13 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 2 0^{\circ} + 13 5^{\circ}

Least Common Multiplier of

9, 4 : 36

9, 4

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

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

9

4

= 3 \cdot 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 3 \cdot 2 \cdot 2 = 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

2 0^{\circ} :

multiply the denominator and numerator by

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

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}

= 2 0^{\circ} + 13 5^{\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} 4 + 486 0 ^{\circ}}{36}

Add similar elements:

72 0^{\circ} + 486 0^{\circ} = 558 0^{\circ}

= 36 0^{\circ} n + 15 5^{\circ}

x = 36 0^{\circ} n + 15 5^{\circ}

x = 36 0^{\circ} n + 15 5^{\circ}

x = 36 0^{\circ} n + 15 5^{\circ}

x = 36 0^{\circ} n + 6 5^{\circ}, x = 36 0^{\circ} n + 15 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-20)= 1/(sqrt(2)) ?
The general solution for sin(x-20)= 1/(sqrt(2)) is x=360n+65,x=360n+155