What is the general solution for sin(2x)=cos(8x) ?

The general solution for sin(2x)=cos(8x) is No Solution for x\in\mathbb{R}

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sin(2x)=cos(8x)

Solution

sin (2 x) = cos (8 x)

Solution

No Solution for x \in R

Solution steps

sin (2 x) = cos (8 x)

Rewrite using trig identities

sin (2 x) = cos (8 x)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (2 x) = sin (\frac{π}{2} - 8 x)

sin (2 x) = sin (\frac{π}{2} - 8 x)

Apply trig inverse properties

sin (2 x) = sin (\frac{π}{2} - 8 x)

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

2 x = \frac{π}{2} - 8 x + 2 πn, 2 x = π - (\frac{π}{2} - 8 x) + 2 πn

2 x = \frac{π}{2} - 8 x + 2 πn, 2 x = π - (\frac{π}{2} - 8 x) + 2 πn

2 x = \frac{π}{2} - 8 x + 2 πn : x = \frac{π + 4 πn}{20}

2 x = \frac{π}{2} - 8 x + 2 πn

Move

8 x

to the left side

2 x = \frac{π}{2} - 8 x + 2 πn

Add

8 x

to both sides

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

Simplify

10 x = \frac{π}{2} + 2 πn

10 x = \frac{π}{2} + 2 πn

Divide both sides by

10

10 x = \frac{π}{2} + 2 πn

Divide both sides by

10

\frac{10 x}{10} = \frac{\frac{π}{2}}{10} + \frac{2 πn}{10}

Simplify

\frac{10 x}{10} = \frac{\frac{π}{2}}{10} + \frac{2 πn}{10}

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{\frac{π}{2}}{10} + \frac{2 πn}{10} : \frac{π + 4 πn}{20}

\frac{\frac{π}{2}}{10} + \frac{2 πn}{10}

Apply rule

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

= \frac{\frac{π}{2} + 2 πn}{10}

Join

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

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{10}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

x = \frac{π + 4 πn}{20}

2 x = π - (\frac{π}{2} - 8 x) + 2 πn : x = - \frac{π + 4 πn}{12}

2 x = π - (\frac{π}{2} - 8 x) + 2 πn

Expand

π - (\frac{π}{2} - 8 x) + 2 πn : π - \frac{π}{2} + 8 x + 2 πn

π - (\frac{π}{2} - 8 x) + 2 πn

- (\frac{π}{2} - 8 x) : - \frac{π}{2} + 8 x

- (\frac{π}{2} - 8 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 8 x)

Apply minus-plus rules

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

= - \frac{π}{2} + 8 x

= π - \frac{π}{2} + 8 x + 2 πn

2 x = π - \frac{π}{2} + 8 x + 2 πn

Move

8 x

to the left side

2 x = π - \frac{π}{2} + 8 x + 2 πn

Subtract

8 x

from both sides

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

Simplify

- 6 x = π - \frac{π}{2} + 2 πn

- 6 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 6

- 6 x = π - \frac{π}{2} + 2 πn

Divide both sides by

- 6

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

Simplify

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

Simplify

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

Apply the fraction rule:

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

= \frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

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

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

Apply rule

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

= \frac{π - \frac{π}{2} + 2 πn}{- 6}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} + 2 πn}{6}

Join

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

π - \frac{π}{2} + 2 πn

Convert element to fraction:

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

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

π 2 - π + 2 πn \cdot 2 = π + 4 πn

π 2 - π + 2 πn \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn

= \frac{π + 4 πn}{2}

= - \frac{\frac{π + 4 πn}{2}}{6}

Simplify

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

\frac{\frac{π + 4 πn}{2}}{6}

Apply the fraction rule:

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

= \frac{π + 4 πn}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π + 4 πn}{12}

= - \frac{π + 4 πn}{12}

x = - \frac{π + 4 πn}{12}

x = - \frac{π + 4 πn}{12}

x = - \frac{π + 4 πn}{12}

Since the equation is undefined for:

\frac{π + 4 πn}{20}, - \frac{π + 4 πn}{12}

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=cos(8x) ?
The general solution for sin(2x)=cos(8x) is No Solution for x\in\mathbb{R}