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

The general solution for cos(x/3)=sin(x) is x=(3pi+12pin)/8 ,x=(3pi+12pin)/4

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cos(x/3)=sin(x)

Solution

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

Solution

x = \frac{3 π + 12 πn}{8}, x = \frac{3 π + 12 πn}{4}

Degrees

x = 67. 5^{\circ} + 27 0^{\circ} n, x = 13 5^{\circ} + 54 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

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

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

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

x = \frac{π}{2} - \frac{x}{3} + 2 πn : x = \frac{3 π + 12 πn}{8}

x = \frac{π}{2} - \frac{x}{3} + 2 πn

Move

\frac{x}{3}

to the left side

x = \frac{π}{2} - \frac{x}{3} + 2 πn

Add

\frac{x}{3}

to both sides

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

Simplify

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

Simplify

x + \frac{x}{3} : \frac{4 x}{3}

x + \frac{x}{3}

Convert element to fraction:

x = \frac{x 3}{3}

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

x + 3 x = 4 x

= \frac{4 x}{3}

Simplify

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

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

Add similar elements:

- \frac{x}{3} + \frac{x}{3} = 0

= \frac{π}{2} + 2 πn

\frac{4 x}{3} = \frac{π}{2} + 2 πn

\frac{4 x}{3} = \frac{π}{2} + 2 πn

\frac{4 x}{3} = \frac{π}{2} + 2 πn

Multiply both sides by

3

\frac{4 x}{3} = \frac{π}{2} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 4 x}{3} : 4 x

\frac{3 \cdot 4 x}{3}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 x}{3}

Divide the numbers:

\frac{12}{3} = 4

= 4 x

Simplify

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

3 \cdot \frac{π}{2} + 3 \cdot 2 πn

Multiply

3 \cdot \frac{π}{2} : \frac{3 π}{2}

3 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 3}{2}

= \frac{3 π}{2} + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= \frac{3 π}{2} + 6 πn

4 x = \frac{3 π}{2} + 6 πn

4 x = \frac{3 π}{2} + 6 πn

4 x = \frac{3 π}{2} + 6 πn

Divide both sides by

4

4 x = \frac{3 π}{2} + 6 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{4} + \frac{6 πn}{4} : \frac{3 π + 12 πn}{8}

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

Apply rule

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

= \frac{\frac{3 π}{2} + 6 πn}{4}

Join

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

\frac{3 π}{2} + 6 πn

Convert element to fraction:

6 πn = \frac{6 πn 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{3 π + 6 πn \cdot 2}{2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{3 π + 12 πn}{2}

= \frac{\frac{3 π + 12 πn}{2}}{4}

Apply the fraction rule:

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

= \frac{3 π + 12 πn}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 π + 12 πn}{8}

x = \frac{3 π + 12 πn}{8}

x = \frac{3 π + 12 πn}{8}

x = \frac{3 π + 12 πn}{8}

x = π - (\frac{π}{2} - \frac{x}{3}) + 2 πn : x = \frac{3 π + 12 πn}{4}

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

Move

(\frac{π}{2} - \frac{x}{3})

to the left side

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

Add

(\frac{π}{2} - \frac{x}{3})

to both sides

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

Simplify

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

Simplify

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

x + \frac{π}{2} - \frac{x}{3}

Convert element to fraction:

x = \frac{x}{1}

= \frac{π}{2} - \frac{x}{3} + \frac{x}{1}

Least Common Multiplier of

2, 3, 1 : 6

2, 3, 1

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

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

2, 3, 1

= 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{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}{1} :

multiply the denominator and numerator by

6

\frac{x}{1} = \frac{x \cdot 6}{1 \cdot 6} = \frac{x \cdot 6}{6}

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

- 2 x + 6 x = 4 x

= \frac{3 π + 4 x}{6}

Simplify

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

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

Add similar elements:

- (\frac{π}{2} - \frac{x}{3}) + \frac{π}{2} - \frac{x}{3} = 0

= π + 2 πn

\frac{3 π + 4 x}{6} = π + 2 πn

\frac{3 π + 4 x}{6} = π + 2 πn

\frac{3 π + 4 x}{6} = π + 2 πn

Multiply both sides by

6

\frac{3 π + 4 x}{6} = π + 2 πn

Multiply both sides by

6

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

Simplify

3 π + 4 x = 6 π + 12 πn

3 π + 4 x = 6 π + 12 πn

Move

3 π

to the right side

3 π + 4 x = 6 π + 12 πn

Subtract

3 π

from both sides

3 π + 4 x - 3 π = 6 π + 12 πn - 3 π

Simplify

4 x = 3 π + 12 πn

4 x = 3 π + 12 πn

Divide both sides by

4

4 x = 3 π + 12 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{3 π}{4} + \frac{12 πn}{4}

Simplify

\frac{4 x}{4} = \frac{3 π}{4} + \frac{12 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

\frac{3 π}{4} + \frac{12 πn}{4}

Apply rule

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

= \frac{3 π + 12 πn}{4}

x = \frac{3 π + 12 πn}{4}

x = \frac{3 π + 12 πn}{4}

x = \frac{3 π + 12 πn}{4}

x = \frac{3 π + 12 πn}{8}, x = \frac{3 π + 12 πn}{4}

x = \frac{3 π + 12 πn}{8}, x = \frac{3 π + 12 πn}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x/3)=sin(x) ?
The general solution for cos(x/3)=sin(x) is x=(3pi+12pin)/8 ,x=(3pi+12pin)/4