What is the general solution for sin(2x+30)=-(sqrt(3))/2 ?

The general solution for sin(2x+30)=-(sqrt(3))/2 is x=180n+105,x=180n+135

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sin(2x+30)=-(sqrt(3))/2

Solution

sin (2 x + 3 0^{\circ}) = - \frac{3}{2}

Solution

x = 18 0^{\circ} n + 10 5^{\circ}, x = 18 0^{\circ} n + 13 5^{\circ}

Radians

x = \frac{7 π}{12} + πn, x = \frac{3 π}{4} + πn

Solution steps

sin (2 x + 3 0^{\circ}) = - \frac{3}{2}

General solutions for

sin (2 x + 3 0^{\circ}) = - \frac{3}{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 + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, 2 x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

2 x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, 2 x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Solve

2 x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 10 5^{\circ}

2 x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

2 x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Subtract

3 0^{\circ}

from both sides

2 x + 3 0^{\circ} - 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Simplify

2 x + 3 0^{\circ} - 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Simplify

2 x + 3 0^{\circ} - 3 0^{\circ} : 2 x

2 x + 3 0^{\circ} - 3 0^{\circ}

Add similar elements:

3 0^{\circ} - 3 0^{\circ} = 0

= 2 x

Simplify

24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 21 0^{\circ}

24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Group like terms

= 36 0^{\circ} n - 3 0^{\circ} + 24 0^{\circ}

Least Common Multiplier of

6, 3 : 6

6, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

24 0^{\circ} :

multiply the denominator and numerator by

2

24 0^{\circ} = \frac{72 0 ^{\circ} 2}{3 \cdot 2} = 24 0^{\circ}

= - 3 0^{\circ} + 24 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} + 144 0 ^{\circ}}{6}

Add similar elements:

- 18 0^{\circ} + 144 0^{\circ} = 126 0^{\circ}

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

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

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{21 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{21 0 ^{\circ}}{2} : 18 0^{\circ} n + 10 5^{\circ}

\frac{36 0 ^{\circ} n}{2} + \frac{21 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{21 0 ^{\circ}}{2} = 10 5^{\circ}

\frac{21 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{126 0 ^{\circ}}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= 10 5^{\circ}

= 18 0^{\circ} n + 10 5^{\circ}

x = 18 0^{\circ} n + 10 5^{\circ}

x = 18 0^{\circ} n + 10 5^{\circ}

x = 18 0^{\circ} n + 10 5^{\circ}

Solve

2 x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n : x = 18 0^{\circ} n + 13 5^{\circ}

2 x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

2 x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Subtract

3 0^{\circ}

from both sides

2 x + 3 0^{\circ} - 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Simplify

2 x + 3 0^{\circ} - 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Simplify

2 x + 3 0^{\circ} - 3 0^{\circ} : 2 x

2 x + 3 0^{\circ} - 3 0^{\circ}

Add similar elements:

3 0^{\circ} - 3 0^{\circ} = 0

= 2 x

Simplify

30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 27 0^{\circ}

30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

Group like terms

= 36 0^{\circ} n - 3 0^{\circ} + 30 0^{\circ}

Least Common Multiplier of

6, 3 : 6

6, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

30 0^{\circ} :

multiply the denominator and numerator by

2

30 0^{\circ} = \frac{90 0 ^{\circ} 2}{3 \cdot 2} = 30 0^{\circ}

= - 3 0^{\circ} + 30 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} + 180 0 ^{\circ}}{6}

Add similar elements:

- 18 0^{\circ} + 180 0^{\circ} = 162 0^{\circ}

= 27 0^{\circ}

Cancel the common factor:

3

= 36 0^{\circ} n + 27 0^{\circ}

2 x = 36 0^{\circ} n + 27 0^{\circ}

2 x = 36 0^{\circ} n + 27 0^{\circ}

2 x = 36 0^{\circ} n + 27 0^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} n + 27 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{27 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{27 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{27 0 ^{\circ}}{2} : 18 0^{\circ} n + 13 5^{\circ}

\frac{36 0 ^{\circ} n}{2} + \frac{27 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{27 0 ^{\circ}}{2} = 13 5^{\circ}

\frac{27 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{54 0 ^{\circ}}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 13 5^{\circ}

= 18 0^{\circ} n + 13 5^{\circ}

x = 18 0^{\circ} n + 13 5^{\circ}

x = 18 0^{\circ} n + 13 5^{\circ}

x = 18 0^{\circ} n + 13 5^{\circ}

x = 18 0^{\circ} n + 10 5^{\circ}, x = 18 0^{\circ} n + 13 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+30)=-(sqrt(3))/2 ?
The general solution for sin(2x+30)=-(sqrt(3))/2 is x=180n+105,x=180n+135