What is the general solution for cos(2x+pi/3)=sin(3x) ?

The general solution for cos(2x+pi/3)=sin(3x) is x=(12pin+pi)/(30),x=pi-pi/6+2pin

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cos(2x+pi/3)=sin(3x)

Solution

cos (2 x + \frac{π}{3}) = sin (3 x)

Solution

x = \frac{12 πn + π}{30}, x = π - \frac{π}{6} + 2 πn

Degrees

x = 6^{\circ} + 7 2^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (2 x + \frac{π}{3}) = sin (3 x)

Rewrite using trig identities

cos (2 x + \frac{π}{3}) = sin (3 x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (2 x + \frac{π}{3}) = sin (\frac{π}{2} - (2 x + \frac{π}{3}))

cos (2 x + \frac{π}{3}) = sin (\frac{π}{2} - (2 x + \frac{π}{3}))

Apply trig inverse properties

cos (2 x + \frac{π}{3}) = sin (\frac{π}{2} - (2 x + \frac{π}{3}))

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

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

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

3 x = \frac{π}{2} - (2 x + \frac{π}{3}) + 2 πn : x = \frac{12 πn + π}{30}

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

Expand

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

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

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

- (2 x + \frac{π}{3})

Distribute parentheses

= - (2 x) - (\frac{π}{3})

Apply minus-plus rules

+ (- a) = - a

= - 2 x - \frac{π}{3}

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

Simplify

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

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

Group like terms

= - 2 x + 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 = π

= - 2 x + 2 πn + \frac{π}{6}

= - 2 x + 2 πn + \frac{π}{6}

3 x = - 2 x + 2 πn + \frac{π}{6}

Move

2 x

to the left side

3 x = - 2 x + 2 πn + \frac{π}{6}

Add

2 x

to both sides

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

Simplify

5 x = 2 πn + \frac{π}{6}

5 x = 2 πn + \frac{π}{6}

Divide both sides by

5

5 x = 2 πn + \frac{π}{6}

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{2 πn}{5} + \frac{\frac{π}{6}}{5} : \frac{12 πn + π}{30}

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

Apply rule

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

= \frac{2 πn + \frac{π}{6}}{5}

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}}{5}

Apply the fraction rule:

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

= \frac{12 πn + π}{6 \cdot 5}

Multiply the numbers:

6 \cdot 5 = 30

= \frac{12 πn + π}{30}

x = \frac{12 πn + π}{30}

x = \frac{12 πn + π}{30}

x = \frac{12 πn + π}{30}

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

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

Expand

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

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

Expand

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

\frac{π}{2} - (2 x + \frac{π}{3})

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

- (2 x + \frac{π}{3})

Distribute parentheses

= - (2 x) - (\frac{π}{3})

Apply minus-plus rules

+ (- a) = - a

= - 2 x - \frac{π}{3}

= \frac{π}{2} - 2 x - \frac{π}{3}

Simplify

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

\frac{π}{2} - 2 x - \frac{π}{3}

Group like terms

= - 2 x + \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 = π

= - 2 x + \frac{π}{6}

= - 2 x + \frac{π}{6}

= π - (- 2 x + \frac{π}{6}) + 2 πn

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

- (- 2 x + \frac{π}{6})

Distribute parentheses

= - (- 2 x) - (\frac{π}{6})

Apply minus-plus rules

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

= 2 x - \frac{π}{6}

= π + 2 x - \frac{π}{6} + 2 πn

3 x = π + 2 x - \frac{π}{6} + 2 πn

Move

2 x

to the left side

3 x = π + 2 x - \frac{π}{6} + 2 πn

Subtract

2 x

from both sides

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

Simplify

x = π - \frac{π}{6} + 2 πn

x = π - \frac{π}{6} + 2 πn

x = \frac{12 πn + π}{30}, x = π - \frac{π}{6} + 2 πn

x = \frac{12 πn + π}{30}, x = π - \frac{π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x+pi/3)=sin(3x) ?
The general solution for cos(2x+pi/3)=sin(3x) is x=(12pin+pi)/(30),x=pi-pi/6+2pin