What is the general solution for sqrt(2)sin(x-20)+1=0 ?

The general solution for sqrt(2)sin(x-20)+1=0 is x=360n+245,x=360n+335

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sqrt(2)sin(x-20)+1=0

Solution

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

Solution

x = 36 0^{\circ} n + 24 5^{\circ}, x = 36 0^{\circ} n + 33 5^{\circ}

Radians

x = \frac{49 π}{36} + 2 πn, x = \frac{67 π}{36} + 2 πn

Solution steps

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

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

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

Cancel the common factor:

2

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

Simplify

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

\frac{- 1}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

Rationalize

- \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}

= - \frac{2}{2}

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

sin (x - 2 0^{\circ}) = - \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} = 22 5^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n

x - 2 0^{\circ} = 22 5^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n

Solve

x - 2 0^{\circ} = 22 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 24 5^{\circ}

x - 2 0^{\circ} = 22 5^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x - 2 0^{\circ} = 22 5^{\circ} + 36 0^{\circ} n

Add

2 0^{\circ}

to both sides

x - 2 0^{\circ} + 2 0^{\circ} = 22 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} = 22 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

22 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 24 5^{\circ}

22 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 2 0^{\circ} + 22 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

22 5^{\circ} :

multiply the denominator and numerator by

9

22 5^{\circ} = \frac{90 0 ^{\circ} 9}{4 \cdot 9} = 22 5^{\circ}

= 2 0^{\circ} + 22 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 + 810 0 ^{\circ}}{36}

Add similar elements:

72 0^{\circ} + 810 0^{\circ} = 882 0^{\circ}

= 36 0^{\circ} n + 24 5^{\circ}

x = 36 0^{\circ} n + 24 5^{\circ}

x = 36 0^{\circ} n + 24 5^{\circ}

x = 36 0^{\circ} n + 24 5^{\circ}

Solve

x - 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 33 5^{\circ}

x - 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x - 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n

Add

2 0^{\circ}

to both sides

x - 2 0^{\circ} + 2 0^{\circ} = 31 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} = 31 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

31 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 33 5^{\circ}

31 5^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 2 0^{\circ} + 31 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

31 5^{\circ} :

multiply the denominator and numerator by

9

31 5^{\circ} = \frac{126 0 ^{\circ} 9}{4 \cdot 9} = 31 5^{\circ}

= 2 0^{\circ} + 31 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 + 1134 0 ^{\circ}}{36}

Add similar elements:

72 0^{\circ} + 1134 0^{\circ} = 1206 0^{\circ}

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

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

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

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

x = 36 0^{\circ} n + 24 5^{\circ}, x = 36 0^{\circ} n + 33 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin(x-20)+1=0 ?
The general solution for sqrt(2)sin(x-20)+1=0 is x=360n+245,x=360n+335