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2sin^2(θ/2-pi/6)-1=0

Solution

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

Solution

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}, θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

Degrees

θ = 15 0^{\circ} + 72 0^{\circ} n, θ = 33 0^{\circ} + 72 0^{\circ} n, θ = 51 0^{\circ} + 72 0^{\circ} n, θ = 69 0^{\circ} + 72 0^{\circ} n

Solution steps

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

Solve by substitution

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

Let:

sin (\frac{θ}{2} - \frac{π}{6}) = u

2 u^{2} - 1 = 0

2 u^{2} - 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

2 u^{2} - 1 = 0

Move

1

to the right side

2 u^{2} - 1 = 0

Add

1

to both sides

2 u^{2} - 1 + 1 = 0 + 1

Simplify

2 u^{2} = 1

2 u^{2} = 1

Divide both sides by

2

2 u^{2} = 1

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{1}{2}

Simplify

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = sin (\frac{θ}{2} - \frac{π}{6})

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}, sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}, sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2} : θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}

General solutions for

sin (\frac{θ}{2} - \frac{π}{6}) = \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}

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

Solve

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn : θ = 4 πn + \frac{5 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn

Add

\frac{π}{6}

to both sides

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

Simplify

\frac{π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{5 π}{12}

\frac{π}{4} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{4} + \frac{π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

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

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

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= \frac{π 3}{12} + \frac{π 2}{12}

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}{12}

Add similar elements:

3 π + 2 π = 5 π

= 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

Multiply both sides by

2

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn + 2 \cdot \frac{5 π}{12} : 4 πn + \frac{5 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{5 π}{12} = \frac{5 π}{6}

2 \cdot \frac{5 π}{12}

Multiply fractions:

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

= \frac{5 π 2}{12}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{12}

Cancel the common factor:

2

= \frac{5 π}{6}

= 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

Solve

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn : θ = 4 πn + \frac{11 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

Add

\frac{π}{6}

to both sides

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{3 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{3 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

Simplify

\frac{3 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{11 π}{12}

\frac{3 π}{4} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{3 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

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

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

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= \frac{π 2}{12} + \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 9 π}{12}

Add similar elements:

2 π + 9 π = 11 π

= 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

Multiply both sides by

2

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn + 2 \cdot \frac{11 π}{12} : 4 πn + \frac{11 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{11 π}{12} = \frac{11 π}{6}

2 \cdot \frac{11 π}{12}

Multiply fractions:

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

= \frac{11 π 2}{12}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{12}

Cancel the common factor:

2

= \frac{11 π}{6}

= 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2} : θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

General solutions for

sin (\frac{θ}{2} - \frac{π}{6}) = - \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}

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

Solve

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn : θ = 4 πn + \frac{17 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn

Add

\frac{π}{6}

to both sides

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{5 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{5 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

Simplify

\frac{5 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{17 π}{12}

\frac{5 π}{4} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{5 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

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

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

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{5 π}{4} :

multiply the denominator and numerator by

3

\frac{5 π}{4} = \frac{5 π 3}{4 \cdot 3} = \frac{15 π}{12}

= \frac{π 2}{12} + \frac{15 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 15 π}{12}

Add similar elements:

2 π + 15 π = 17 π

= 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

Multiply both sides by

2

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12} : 4 πn + \frac{17 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{17 π}{12} = \frac{17 π}{6}

2 \cdot \frac{17 π}{12}

Multiply fractions:

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

= \frac{17 π 2}{12}

Multiply the numbers:

17 \cdot 2 = 34

= \frac{34 π}{12}

Cancel the common factor:

2

= \frac{17 π}{6}

= 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

Solve

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn : θ = 4 πn + \frac{23 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

Add

\frac{π}{6}

to both sides

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{4} + 2 πn + \frac{π}{6}

Simplify

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

Simplify

\frac{7 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{23 π}{12}

\frac{7 π}{4} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{6} + \frac{7 π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

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

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

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{7 π}{4} :

multiply the denominator and numerator by

3

\frac{7 π}{4} = \frac{7 π 3}{4 \cdot 3} = \frac{21 π}{12}

= \frac{π 2}{12} + \frac{21 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 21 π}{12}

Add similar elements:

2 π + 21 π = 23 π

= 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

Multiply both sides by

2

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

Simplify

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot 2 πn + 2 \cdot \frac{23 π}{12} : 4 πn + \frac{23 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{23 π}{12} = \frac{23 π}{6}

2 \cdot \frac{23 π}{12}

Multiply fractions:

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

= \frac{23 π 2}{12}

Multiply the numbers:

23 \cdot 2 = 46

= \frac{46 π}{12}

Cancel the common factor:

2

= \frac{23 π}{6}

= 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

Combine all the solutions

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}, θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

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