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Popular Trigonometry >

sin((5pi)/6-2x)=cos(x-pi/6),sin((2pi)/3-x)

Solution

sin (\frac{5 π}{6} - 2 x) = cos (x - \frac{π}{6}), sin (\frac{2 π}{3} - x)

Solution

No Solution for x \in R

Solution steps

sin (\frac{5 π}{6} - 2 x) = cos (x - \frac{π}{6}), sin (\frac{2 π}{3} - x)

Rewrite using trig identities

sin (\frac{5 π}{6} - 2 x) = cos (x - \frac{π}{6})

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (\frac{5 π}{6} - 2 x) = sin (\frac{π}{2} - (x - \frac{π}{6}))

sin (\frac{5 π}{6} - 2 x) = sin (\frac{π}{2} - (x - \frac{π}{6}))

Apply trig inverse properties

sin (\frac{5 π}{6} - 2 x) = sin (\frac{π}{2} - (x - \frac{π}{6}))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

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

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

\frac{5 π}{6} - 2 x = \frac{π}{2} - (x - \frac{π}{6}) + 2 πn : x = - \frac{12 πn - π}{6}

\frac{5 π}{6} - 2 x = \frac{π}{2} - (x - \frac{π}{6}) + 2 πn

Expand

\frac{π}{2} - (x - \frac{π}{6}) + 2 πn : - x + 2 πn + \frac{2 π}{3}

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

- (x - \frac{π}{6}) : - x + \frac{π}{6}

- (x - \frac{π}{6})

Distribute parentheses

= - (x) - (- \frac{π}{6})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - x + \frac{π}{6}

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

Simplify

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

\frac{π}{2} - x + \frac{π}{6} + 2 πn

Group like terms

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

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 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

2

6

= 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}

= \frac{π 3}{6} + \frac{π}{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}{6}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{6}

Cancel the common factor:

2

= - x + 2 πn + \frac{2 π}{3}

= - x + 2 πn + \frac{2 π}{3}

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

Move

\frac{5 π}{6}

to the right side

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

Subtract

\frac{5 π}{6}

from both sides

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

Simplify

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

Simplify

\frac{5 π}{6} - 2 x - \frac{5 π}{6} : - 2 x

\frac{5 π}{6} - 2 x - \frac{5 π}{6}

Add similar elements:

\frac{5 π}{6} - \frac{5 π}{6} = 0

= - 2 x

Simplify

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

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

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= \frac{4 π}{6} - \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π - 5 π}{6}

Add similar elements:

4 π - 5 π = - π

= \frac{- π}{6}

Apply the fraction rule:

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

= - x + 2 πn - \frac{π}{6}

- 2 x = - x + 2 πn - \frac{π}{6}

- 2 x = - x + 2 πn - \frac{π}{6}

- 2 x = - x + 2 πn - \frac{π}{6}

Move

x

to the left side

- 2 x = - x + 2 πn - \frac{π}{6}

Add

x

to both sides

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

Simplify

- x = 2 πn - \frac{π}{6}

- x = 2 πn - \frac{π}{6}

Divide both sides by

- 1

- x = 2 πn - \frac{π}{6}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{\frac{π}{6}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{\frac{π}{6}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 πn}{- 1} - \frac{\frac{π}{6}}{- 1} : - \frac{12 πn - π}{6}

\frac{2 πn}{- 1} - \frac{\frac{π}{6}}{- 1}

Apply rule

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

= \frac{2 πn - \frac{π}{6}}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn - \frac{π}{6}}{1}

Join

2 πn - \frac{π}{6} : \frac{12 πn - π}{6}

2 πn - \frac{π}{6}

Convert element to fraction:

2 πn = \frac{2 πn 6}{6}

= \frac{2 πn \cdot 6}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 πn \cdot 6 - π}{6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{12 πn - π}{6}

= - \frac{\frac{12 πn - π}{6}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{12 πn - π}{6}

x = - \frac{12 πn - π}{6}

x = - \frac{12 πn - π}{6}

x = - \frac{12 πn - π}{6}

\frac{5 π}{6} - 2 x = π - (\frac{π}{2} - (x - \frac{π}{6})) + 2 πn : x = - \frac{- π + 4 πn}{6}

\frac{5 π}{6} - 2 x = π - (\frac{π}{2} - (x - \frac{π}{6})) + 2 πn

Expand

π - (\frac{π}{2} - (x - \frac{π}{6})) + 2 πn : π + x - \frac{2 π}{3} + 2 πn

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

Expand

\frac{π}{2} - (x - \frac{π}{6}) : - x + \frac{2 π}{3}

\frac{π}{2} - (x - \frac{π}{6})

- (x - \frac{π}{6}) : - x + \frac{π}{6}

- (x - \frac{π}{6})

Distribute parentheses

= - (x) - (- \frac{π}{6})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - x + \frac{π}{6}

= \frac{π}{2} - x + \frac{π}{6}

Simplify

\frac{π}{2} - x + \frac{π}{6} : - x + \frac{2 π}{3}

\frac{π}{2} - x + \frac{π}{6}

Group like terms

= - x + \frac{π}{2} + \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 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

2

6

= 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}

= \frac{π 3}{6} + \frac{π}{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}{6}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{6}

Cancel the common factor:

2

= - x + \frac{2 π}{3}

= - x + \frac{2 π}{3}

= π - (- x + \frac{2 π}{3}) + 2 πn

- (- x + \frac{2 π}{3}) : x - \frac{2 π}{3}

- (- x + \frac{2 π}{3})

Distribute parentheses

= - (- x) - (\frac{2 π}{3})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= x - \frac{2 π}{3}

= π + x - \frac{2 π}{3} + 2 πn

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

Move

\frac{5 π}{6}

to the right side

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

Subtract

\frac{5 π}{6}

from both sides

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

Simplify

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

Simplify

\frac{5 π}{6} - 2 x - \frac{5 π}{6} : - 2 x

\frac{5 π}{6} - 2 x - \frac{5 π}{6}

Add similar elements:

\frac{5 π}{6} - \frac{5 π}{6} = 0

= - 2 x

Simplify

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

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

Group like terms

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

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= - \frac{4 π}{6} - \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 π - 5 π}{6}

Add similar elements:

- 4 π - 5 π = - 9 π

= \frac{- 9 π}{6}

Apply the fraction rule:

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

= - \frac{9 π}{6}

Cancel the common factor:

3

= x + π + 2 πn - \frac{3 π}{2}

- 2 x = x + π + 2 πn - \frac{3 π}{2}

- 2 x = x + π + 2 πn - \frac{3 π}{2}

- 2 x = x + π + 2 πn - \frac{3 π}{2}

Move

x

to the left side

- 2 x = x + π + 2 πn - \frac{3 π}{2}

Subtract

x

from both sides

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

Simplify

- 3 x = π + 2 πn - \frac{3 π}{2}

- 3 x = π + 2 πn - \frac{3 π}{2}

Divide both sides by

- 3

- 3 x = π + 2 πn - \frac{3 π}{2}

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

Apply rule

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

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

Apply the fraction rule:

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

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

Join

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

π + 2 πn - \frac{3 π}{2}

Convert element to fraction:

π = \frac{π 2}{2}, 2 πn = \frac{2 πn 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 2 πn \cdot 2 - 3 π}{2}

π 2 + 2 πn \cdot 2 - 3 π = - π + 4 πn

π 2 + 2 πn \cdot 2 - 3 π

Add similar elements:

2 π - 3 π = - π

= - π + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= - π + 4 πn

= \frac{- π + 4 πn}{2}

= - \frac{\frac{4 πn - π}{2}}{3}

Simplify

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

\frac{\frac{- π + 4 πn}{2}}{3}

Apply the fraction rule:

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

= \frac{- π + 4 πn}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- π + 4 πn}{6}

= - \frac{4 πn - π}{6}

= - \frac{- π + 4 πn}{6}

x = - \frac{- π + 4 πn}{6}

x = - \frac{- π + 4 πn}{6}

x = - \frac{- π + 4 πn}{6}

Solutions for the range

sin (\frac{2 π}{3} - x)

No Solution for x \in R

Graph

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