What is the general solution for 2sin(x+pi/3)=-1 ?

The general solution for 2sin(x+pi/3)=-1 is x=2pin+(5pi)/6 ,x=2pin+(3pi)/2

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2sin(x+pi/3)=-1

Solution

2 sin (x + \frac{π}{3}) = - 1

Solution

x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{3 π}{2}

Degrees

x = 15 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x + \frac{π}{3}) = - 1

Divide both sides by

2

2 sin (x + \frac{π}{3}) = - 1

Divide both sides by

2

\frac{2 sin ( x + \frac{π}{3} )}{2} = \frac{- 1}{2}

Simplify

sin (x + \frac{π}{3}) = - \frac{1}{2}

sin (x + \frac{π}{3}) = - \frac{1}{2}

General solutions for

sin (x + \frac{π}{3}) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x + \frac{π}{3} = \frac{7 π}{6} + 2 πn, x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

x + \frac{π}{3} = \frac{7 π}{6} + 2 πn, x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Solve

x + \frac{π}{3} = \frac{7 π}{6} + 2 πn : x = 2 πn + \frac{5 π}{6}

x + \frac{π}{3} = \frac{7 π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

x + \frac{π}{3} = \frac{7 π}{6} + 2 πn

Subtract

\frac{π}{3}

from both sides

x + \frac{π}{3} - \frac{π}{3} = \frac{7 π}{6} + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} = \frac{7 π}{6} + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} : x

x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= x

Simplify

\frac{7 π}{6} + 2 πn - \frac{π}{3} : 2 πn + \frac{5 π}{6}

\frac{7 π}{6} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{7 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= - \frac{π 2}{6} + \frac{7 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + 7 π}{6}

Add similar elements:

- 2 π + 7 π = 5 π

= 2 πn + \frac{5 π}{6}

x = 2 πn + \frac{5 π}{6}

x = 2 πn + \frac{5 π}{6}

x = 2 πn + \frac{5 π}{6}

Solve

x + \frac{π}{3} = \frac{11 π}{6} + 2 πn : x = 2 πn + \frac{3 π}{2}

x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Subtract

\frac{π}{3}

from both sides

x + \frac{π}{3} - \frac{π}{3} = \frac{11 π}{6} + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} = \frac{11 π}{6} + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} : x

x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= x

Simplify

\frac{11 π}{6} + 2 πn - \frac{π}{3} : 2 πn + \frac{3 π}{2}

\frac{11 π}{6} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{11 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= - \frac{π 2}{6} + \frac{11 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + 11 π}{6}

Add similar elements:

- 2 π + 11 π = 9 π

= \frac{9 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{3 π}{2}

Popular Examples

cot^2(a)=cos^2(a)+cos(a)cos(a)

cot^{2} (a) = cos^{2} (a) + cos (a) cos (a)

cot(x)*tan(2x)=3

cot (x) \cdot tan (2 x) = 3

1*sin(35)=1.33*sin(θ)

1 \cdot sin (3 5^{\circ}) = 1.33 \cdot sin (θ)

cos(3A)=1

cos (3 A) = 1

cos(θ)=(-0.68159)

cos (θ) = (- 0.68159)

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x+pi/3)=-1 ?
The general solution for 2sin(x+pi/3)=-1 is x=2pin+(5pi)/6 ,x=2pin+(3pi)/2