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Popular Trigonometry >

cos^2(2x+pi/6)= 1/2

Solution

cos^{2} (2 x + \frac{π}{6}) = \frac{1}{2}

Solution

x = πn + \frac{π}{24}, x = π + πn - \frac{5 π}{24}, x = πn + \frac{7 π}{24}, x = πn - \frac{11 π}{24}

Degrees

x = 7. 5^{\circ} + 18 0^{\circ} n, x = 142. 5^{\circ} + 18 0^{\circ} n, x = 52. 5^{\circ} + 18 0^{\circ} n, x = - 82. 5^{\circ} + 18 0^{\circ} n

Solution steps

cos^{2} (2 x + \frac{π}{6}) = \frac{1}{2}

Solve by substitution

cos^{2} (2 x + \frac{π}{6}) = \frac{1}{2}

Let:

cos (2 x + \frac{π}{6}) = u

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

u^{2} = \frac{1}{2} : u = \frac{1}{2}, u = - \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 = cos (2 x + \frac{π}{6})

cos (2 x + \frac{π}{6}) = \frac{1}{2}, cos (2 x + \frac{π}{6}) = - \frac{1}{2}

cos (2 x + \frac{π}{6}) = \frac{1}{2}, cos (2 x + \frac{π}{6}) = - \frac{1}{2}

cos (2 x + \frac{π}{6}) = \frac{1}{2} : x = πn + \frac{π}{24}, x = π + πn - \frac{5 π}{24}

cos (2 x + \frac{π}{6}) = \frac{1}{2}

Apply trig inverse properties

cos (2 x + \frac{π}{6}) = \frac{1}{2}

General solutions for

cos (2 x + \frac{π}{6}) = \frac{1}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x + \frac{π}{6} = arccos (\frac{1}{2}) + 2 πn, 2 x + \frac{π}{6} = 2 π - arccos (\frac{1}{2}) + 2 πn

2 x + \frac{π}{6} = arccos (\frac{1}{2}) + 2 πn, 2 x + \frac{π}{6} = 2 π - arccos (\frac{1}{2}) + 2 πn

Solve

2 x + \frac{π}{6} = arccos (\frac{1}{2}) + 2 πn : x = πn + \frac{π}{24}

2 x + \frac{π}{6} = arccos (\frac{1}{2}) + 2 πn

Simplify

arccos (\frac{1}{2}) + 2 πn : \frac{π}{4} + 2 πn

arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{4} + 2 πn

2 x + \frac{π}{6} = \frac{π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

2 x + \frac{π}{6} = \frac{π}{4} + 2 πn

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

Simplify

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

= 2 πn + \frac{π}{12}

2 x = 2 πn + \frac{π}{12}

2 x = 2 πn + \frac{π}{12}

2 x = 2 πn + \frac{π}{12}

Divide both sides by

2

2 x = 2 πn + \frac{π}{12}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{π}{12}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{π}{12}}{2} : πn + \frac{π}{24}

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{12}}{2} = \frac{π}{24}

\frac{\frac{π}{12}}{2}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{π}{24}

= πn + \frac{π}{24}

x = πn + \frac{π}{24}

x = πn + \frac{π}{24}

x = πn + \frac{π}{24}

Solve

2 x + \frac{π}{6} = 2 π - arccos (\frac{1}{2}) + 2 πn : x = π + πn - \frac{5 π}{24}

2 x + \frac{π}{6} = 2 π - arccos (\frac{1}{2}) + 2 πn

Simplify

2 π - arccos (\frac{1}{2}) + 2 πn : 2 π - \frac{π}{4} + 2 πn

2 π - arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{4} + 2 πn

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

Move

\frac{π}{6}

to the right side

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

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

Simplify

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

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

Group like terms

= 2 π + 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

= \frac{- 5 π}{12}

Apply the fraction rule:

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

= 2 π + 2 πn - \frac{5 π}{12}

2 x = 2 π + 2 πn - \frac{5 π}{12}

2 x = 2 π + 2 πn - \frac{5 π}{12}

2 x = 2 π + 2 πn - \frac{5 π}{12}

Divide both sides by

2

2 x = 2 π + 2 πn - \frac{5 π}{12}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{12}}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{12}}{2} : π + πn - \frac{5 π}{24}

\frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{12}}{2}

\frac{2 π}{2} = π

\frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{5 π}{12}}{2} = \frac{5 π}{24}

\frac{\frac{5 π}{12}}{2}

Apply the fraction rule:

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

= \frac{5 π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{5 π}{24}

= π + πn - \frac{5 π}{24}

x = π + πn - \frac{5 π}{24}

x = π + πn - \frac{5 π}{24}

x = π + πn - \frac{5 π}{24}

x = πn + \frac{π}{24}, x = π + πn - \frac{5 π}{24}

cos (2 x + \frac{π}{6}) = - \frac{1}{2} : x = πn + \frac{7 π}{24}, x = πn - \frac{11 π}{24}

cos (2 x + \frac{π}{6}) = - \frac{1}{2}

Apply trig inverse properties

cos (2 x + \frac{π}{6}) = - \frac{1}{2}

General solutions for

cos (2 x + \frac{π}{6}) = - \frac{1}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

2 x + \frac{π}{6} = arccos (- \frac{1}{2}) + 2 πn, 2 x + \frac{π}{6} = - arccos (- \frac{1}{2}) + 2 πn

2 x + \frac{π}{6} = arccos (- \frac{1}{2}) + 2 πn, 2 x + \frac{π}{6} = - arccos (- \frac{1}{2}) + 2 πn

Solve

2 x + \frac{π}{6} = arccos (- \frac{1}{2}) + 2 πn : x = πn + \frac{7 π}{24}

2 x + \frac{π}{6} = arccos (- \frac{1}{2}) + 2 πn

Simplify

arccos (- \frac{1}{2}) + 2 πn : \frac{3 π}{4} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{3 π}{4} + 2 πn

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

Move

\frac{π}{6}

to the right side

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

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

Simplify

\frac{3 π}{4} + 2 πn - \frac{π}{6} : 2 πn + \frac{7 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 = 7 π

= 2 πn + \frac{7 π}{12}

2 x = 2 πn + \frac{7 π}{12}

2 x = 2 πn + \frac{7 π}{12}

2 x = 2 πn + \frac{7 π}{12}

Divide both sides by

2

2 x = 2 πn + \frac{7 π}{12}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{7 π}{12}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{7 π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{7 π}{12}}{2} : πn + \frac{7 π}{24}

\frac{2 πn}{2} + \frac{\frac{7 π}{12}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{7 π}{12}}{2} = \frac{7 π}{24}

\frac{\frac{7 π}{12}}{2}

Apply the fraction rule:

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

= \frac{7 π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{7 π}{24}

= πn + \frac{7 π}{24}

x = πn + \frac{7 π}{24}

x = πn + \frac{7 π}{24}

x = πn + \frac{7 π}{24}

Solve

2 x + \frac{π}{6} = - arccos (- \frac{1}{2}) + 2 πn : x = πn - \frac{11 π}{24}

2 x + \frac{π}{6} = - arccos (- \frac{1}{2}) + 2 πn

Simplify

- arccos (- \frac{1}{2}) + 2 πn : - \frac{3 π}{4} + 2 πn

- arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= - \frac{3 π}{4} + 2 πn

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

Move

\frac{π}{6}

to the right side

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

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

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

= \frac{- 11 π}{12}

Apply the fraction rule:

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

= 2 πn - \frac{11 π}{12}

2 x = 2 πn - \frac{11 π}{12}

2 x = 2 πn - \frac{11 π}{12}

2 x = 2 πn - \frac{11 π}{12}

Divide both sides by

2

2 x = 2 πn - \frac{11 π}{12}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} - \frac{\frac{11 π}{12}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} - \frac{\frac{11 π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} - \frac{\frac{11 π}{12}}{2} : πn - \frac{11 π}{24}

\frac{2 πn}{2} - \frac{\frac{11 π}{12}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{11 π}{12}}{2} = \frac{11 π}{24}

\frac{\frac{11 π}{12}}{2}

Apply the fraction rule:

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

= \frac{11 π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{11 π}{24}

= πn - \frac{11 π}{24}

x = πn - \frac{11 π}{24}

x = πn - \frac{11 π}{24}

x = πn - \frac{11 π}{24}

x = πn + \frac{7 π}{24}, x = πn - \frac{11 π}{24}

Combine all the solutions

x = πn + \frac{π}{24}, x = π + πn - \frac{5 π}{24}, x = πn + \frac{7 π}{24}, x = πn - \frac{11 π}{24}

Graph

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