What is the general solution for cos(3x)=-sin(x+pi/6) ?

The general solution for cos(3x)=-sin(x+pi/6) is x=(pi(3n+1))/3 ,x=-(pi(3n+1))/6

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cos(3x)=-sin(x+pi/6)

Solution

cos (3 x) = - sin (x + \frac{π}{6})

Solution

x = \frac{π ( 3 n + 1 )}{3}, x = - \frac{π ( 3 n + 1 )}{6}

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} - 9 0^{\circ} n

Solution steps

cos (3 x) = - sin (x + \frac{π}{6})

Rewrite using trig identities

cos (3 x) = - sin (x + \frac{π}{6})

Use the following identity:

- sin (x) = sin (- x)

cos (3 x) = sin (- (x + \frac{π}{6}))

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (3 x) = sin (\frac{π}{2} - 3 x)

cos (3 x) = sin (\frac{π}{2} - 3 x)

Apply trig inverse properties

cos (3 x) = sin (\frac{π}{2} - 3 x)

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

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

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

- (x + \frac{π}{6}) = \frac{π}{2} - 3 x + 2 πn : x = \frac{π ( 3 n + 1 )}{3}

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

Expand

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

- (x + \frac{π}{6})

Distribute parentheses

= - (x) - (\frac{π}{6})

Apply minus-plus rules

+ (- a) = - a

= - x - \frac{π}{6}

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

Move

\frac{π}{6}

to the right side

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

Add

\frac{π}{6}

to both sides

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

Simplify

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

Simplify

- x - \frac{π}{6} + \frac{π}{6} : - x

- x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= - x

Simplify

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

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

Group like terms

= - 3 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

= - 3 x + 2 πn + \frac{2 π}{3}

- x = - 3 x + 2 πn + \frac{2 π}{3}

- x = - 3 x + 2 πn + \frac{2 π}{3}

- x = - 3 x + 2 πn + \frac{2 π}{3}

Move

3 x

to the left side

- x = - 3 x + 2 πn + \frac{2 π}{3}

Add

3 x

to both sides

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

Simplify

2 x = 2 πn + \frac{2 π}{3}

2 x = 2 πn + \frac{2 π}{3}

Divide both sides by

2

2 x = 2 πn + \frac{2 π}{3}

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{2 π}{3}}{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{2 π}{3}}{2} : \frac{π ( 3 n + 1 )}{3}

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

Apply rule

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

= \frac{2 πn + \frac{2 π}{3}}{2}

Join

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

2 πn + \frac{2 π}{3}

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 πn \cdot 3 + 2 π}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 πn + 2 π}{3}

= \frac{\frac{6 πn + 2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{6 πn + 2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 πn + 2 π}{6}

Factor

6 πn + 2 π : 2 π (3 n + 1)

6 πn + 2 π

Rewrite as

= 3 \cdot 2 πn + 1 \cdot 2 π

Factor out common term

2 π

= 2 π (3 n + 1)

= \frac{2 π ( 3 n + 1 )}{6}

Cancel the common factor:

2

= \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

x = \frac{π ( 3 n + 1 )}{3}

- (x + \frac{π}{6}) = π - (\frac{π}{2} - 3 x) + 2 πn : x = - \frac{π ( 3 n + 1 )}{6}

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

Expand

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

- (x + \frac{π}{6})

Distribute parentheses

= - (x) - (\frac{π}{6})

Apply minus-plus rules

+ (- a) = - a

= - x - \frac{π}{6}

Expand

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

π - (\frac{π}{2} - 3 x) + 2 πn

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

- (\frac{π}{2} - 3 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x

= π - \frac{π}{2} + 3 x + 2 πn

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

Move

\frac{π}{6}

to the right side

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

Add

\frac{π}{6}

to both sides

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

Simplify

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

Simplify

- x - \frac{π}{6} + \frac{π}{6} : - x

- x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} = 0

= - x

Simplify

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

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

Group like terms

= 3 x + π + 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 = - 2 π

= \frac{- 2 π}{6}

Apply the fraction rule:

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

= - \frac{2 π}{6}

Cancel the common factor:

2

= 3 x + π + 2 πn - \frac{π}{3}

- x = 3 x + π + 2 πn - \frac{π}{3}

- x = 3 x + π + 2 πn - \frac{π}{3}

- x = 3 x + π + 2 πn - \frac{π}{3}

Move

3 x

to the left side

- x = 3 x + π + 2 πn - \frac{π}{3}

Subtract

3 x

from both sides

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

Simplify

- 4 x = π + 2 πn - \frac{π}{3}

- 4 x = π + 2 πn - \frac{π}{3}

Divide both sides by

- 4

- 4 x = π + 2 πn - \frac{π}{3}

Divide both sides by

- 4

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

Simplify

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

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{π}{- 4} + \frac{2 πn}{- 4} - \frac{\frac{π}{3}}{- 4} : - \frac{π ( 3 n + 1 )}{6}

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

Apply rule

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

= \frac{π + 2 πn - \frac{π}{3}}{- 4}

Apply the fraction rule:

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

= - \frac{π + 2 πn - \frac{π}{3}}{4}

Join

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

π + 2 πn - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}, 2 πn = \frac{2 πn 3}{3}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 2 πn \cdot 3 - π}{3}

π 3 + 2 πn \cdot 3 - π = 2 π + 6 πn

π 3 + 2 πn \cdot 3 - π

Add similar elements:

3 π - π = 2 π

= 2 π + 2 \cdot 3 πn

Multiply the numbers:

2 \cdot 3 = 6

= 2 π + 6 πn

= \frac{2 π + 6 πn}{3}

= - \frac{\frac{2 π + 6 πn}{3}}{4}

Simplify

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

\frac{\frac{2 π + 6 πn}{3}}{4}

Apply the fraction rule:

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

= \frac{2 π + 6 πn}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{2 π + 6 πn}{12}

= - \frac{2 π + 6 πn}{12}

Cancel

\frac{2 π + 6 πn}{12} : \frac{π ( 3 n + 1 )}{6}

\frac{2 π + 6 πn}{12}

Factor

2 π + 6 πn : 2 π (1 + 3 n)

2 π + 6 πn

Rewrite as

= 1 \cdot 2 π + 3 \cdot 2 πn

Factor out common term

2 π

= 2 π (1 + 3 n)

= \frac{2 π ( 1 + 3 n )}{12}

Cancel the common factor:

2

= \frac{π ( 3 n + 1 )}{6}

= - \frac{π ( 3 n + 1 )}{6}

x = - \frac{π ( 3 n + 1 )}{6}

x = - \frac{π ( 3 n + 1 )}{6}

x = - \frac{π ( 3 n + 1 )}{6}

x = \frac{π ( 3 n + 1 )}{3}, x = - \frac{π ( 3 n + 1 )}{6}

x = \frac{π ( 3 n + 1 )}{3}, x = - \frac{π ( 3 n + 1 )}{6}

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Frequently Asked Questions (FAQ)

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