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

The general solution for sin(x/3)=cos(x/2) is x=(3pi+12pin)/5 ,x=-3pi-12pin

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

Solution

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

Solution

x = \frac{3 π + 12 πn}{5}, x = - 3 π - 12 πn

Degrees

x = 10 8^{\circ} + 43 2^{\circ} n, x = - 54 0^{\circ} - 216 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

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

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

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

\frac{x}{3} = \frac{π}{2} - \frac{x}{2} + 2 πn : x = \frac{3 π + 12 πn}{5}

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

Move

\frac{x}{2}

to the left side

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

Add

\frac{x}{2}

to both sides

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

Simplify

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

Simplify

\frac{x}{3} + \frac{x}{2} : \frac{5 x}{6}

\frac{x}{3} + \frac{x}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 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{x}{3} :

multiply the denominator and numerator by

2

\frac{x}{3} = \frac{x \cdot 2}{3 \cdot 2} = \frac{x \cdot 2}{6}

For

\frac{x}{2} :

multiply the denominator and numerator by

3

\frac{x}{2} = \frac{x \cdot 3}{2 \cdot 3} = \frac{x \cdot 3}{6}

= \frac{x \cdot 2}{6} + \frac{x \cdot 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 2 + x \cdot 3}{6}

Add similar elements:

2 x + 3 x = 5 x

= \frac{5 x}{6}

Simplify

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

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

Add similar elements:

- \frac{x}{2} + \frac{x}{2} = 0

= \frac{π}{2} + 2 πn

\frac{5 x}{6} = \frac{π}{2} + 2 πn

\frac{5 x}{6} = \frac{π}{2} + 2 πn

\frac{5 x}{6} = \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{5 x}{6} = \frac{π}{2} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

\frac{6 \cdot 5 x}{6} : 5 x

\frac{6 \cdot 5 x}{6}

Multiply the numbers:

6 \cdot 5 = 30

= \frac{30 x}{6}

Divide the numbers:

\frac{30}{6} = 5

= 5 x

Simplify

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

6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 3 π + 12 πn

5 x = 3 π + 12 πn

5 x = 3 π + 12 πn

5 x = 3 π + 12 πn

Divide both sides by

5

5 x = 3 π + 12 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{12 πn}{5}

Simplify

\frac{5 x}{5} = \frac{3 π}{5} + \frac{12 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{3 π}{5} + \frac{12 πn}{5} : \frac{3 π + 12 πn}{5}

\frac{3 π}{5} + \frac{12 πn}{5}

Apply rule

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

= \frac{3 π + 12 πn}{5}

x = \frac{3 π + 12 πn}{5}

x = \frac{3 π + 12 πn}{5}

x = \frac{3 π + 12 πn}{5}

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

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

Expand

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

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

Apply rule

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

= π - \frac{- x + π}{2} + 2 πn

Apply the fraction rule:

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

\frac{π - x}{2} = - (\frac{π}{2}) - (- \frac{x}{2})

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

Remove parentheses:

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

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

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

Multiply by LCM

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

Find Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Multiply by LCM=

6

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

Simplify

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

Simplify

\frac{x}{3} \cdot 6 : 2 x

\frac{x}{3} \cdot 6

Multiply fractions:

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

= \frac{x \cdot 6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 x

Simplify

π 6 : 6 π

π 6

Apply the commutative law:

π 6 = 6 π

6 π

Simplify

- \frac{π}{2} \cdot 6 : - 3 π

- \frac{π}{2} \cdot 6

Multiply fractions:

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

= - \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3 π

Simplify

\frac{x}{2} \cdot 6 : 3 x

\frac{x}{2} \cdot 6

Multiply fractions:

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

= \frac{x \cdot 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 πn \cdot 6 : 12 πn

2 πn \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 12 πn

2 x = 6 π - 3 π + 3 x + 12 πn

2 x = 3 π + 3 x + 12 πn

2 x = 3 π + 3 x + 12 πn

2 x = 3 π + 3 x + 12 πn

Move

3 x

to the left side

2 x = 3 π + 3 x + 12 πn

Subtract

3 x

from both sides

2 x - 3 x = 3 π + 3 x + 12 πn - 3 x

Simplify

- x = 3 π + 12 πn

- x = 3 π + 12 πn

Divide both sides by

- 1

- x = 3 π + 12 πn

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{3 π}{- 1} + \frac{12 πn}{- 1}

Simplify

\frac{- x}{- 1} = \frac{3 π}{- 1} + \frac{12 πn}{- 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{3 π}{- 1} + \frac{12 πn}{- 1} : - 3 π - 12 πn

\frac{3 π}{- 1} + \frac{12 πn}{- 1}

Apply rule

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

= \frac{3 π + 12 πn}{- 1}

Apply the fraction rule:

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

= - \frac{3 π + 12 πn}{1}

Apply rule

\frac{a}{1} = a

= - (3 π + 12 πn)

Distribute parentheses

= - (3 π) - (12 πn)

Apply minus-plus rules

+ (- a) = - a

= - 3 π - 12 πn

x = - 3 π - 12 πn

x = - 3 π - 12 πn

x = - 3 π - 12 πn

x = \frac{3 π + 12 πn}{5}, x = - 3 π - 12 πn

x = \frac{3 π + 12 πn}{5}, x = - 3 π - 12 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x/3)=cos(x/2) ?
The general solution for sin(x/3)=cos(x/2) is x=(3pi+12pin)/5 ,x=-3pi-12pin