What is the general solution for sin(θ-pi/3)-1=0 ?

The general solution for sin(θ-pi/3)-1=0 is θ=2pin+(5pi)/6

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sin(θ-pi/3)-1=0

Solution

sin (θ - \frac{π}{3}) - 1 = 0

Solution

θ = 2 πn + \frac{5 π}{6}

Degrees

θ = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ - \frac{π}{3}) - 1 = 0

Move

1

to the right side

sin (θ - \frac{π}{3}) - 1 = 0

Add

1

to both sides

sin (θ - \frac{π}{3}) - 1 + 1 = 0 + 1

Simplify

sin (θ - \frac{π}{3}) = 1

sin (θ - \frac{π}{3}) = 1

General solutions for

sin (θ - \frac{π}{3}) = 1

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}

θ - \frac{π}{3} = \frac{π}{2} + 2 πn

θ - \frac{π}{3} = \frac{π}{2} + 2 πn

Solve

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

θ - \frac{π}{3} = \frac{π}{2} + 2 πn

Move

\frac{π}{3}

to the right side

θ - \frac{π}{3} = \frac{π}{2} + 2 πn

Add

\frac{π}{3}

to both sides

θ - \frac{π}{3} + \frac{π}{3} = \frac{π}{2} + 2 πn + \frac{π}{3}

Simplify

θ - \frac{π}{3} + \frac{π}{3} = \frac{π}{2} + 2 πn + \frac{π}{3}

Simplify

θ - \frac{π}{3} + \frac{π}{3} : θ

θ - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= θ

Simplify

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

\frac{π}{2} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

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

\frac{π}{2} :

multiply the denominator and numerator by

3

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

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π 2}{6}

Add similar elements:

3 π + 2 π = 5 π

= 2 πn + \frac{5 π}{6}

θ = 2 πn + \frac{5 π}{6}

θ = 2 πn + \frac{5 π}{6}

θ = 2 πn + \frac{5 π}{6}

θ = 2 πn + \frac{5 π}{6}

Popular Examples

cos(5x)= 1/2

cos (5 x) = \frac{1}{2}

sin(θ)= 3/5 cos(θ)

sin (θ) = \frac{3}{5} cos (θ)

cos(x/3)=0

cos (\frac{x}{3}) = 0

sin(a)= 5/6

sin (a) = \frac{5}{6}

2sin^2(x)cos(x)=cos(x)

2 sin^{2} (x) cos (x) = cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for sin(θ-pi/3)-1=0 ?
The general solution for sin(θ-pi/3)-1=0 is θ=2pin+(5pi)/6