What is the general solution for sin(4x)=cos(x/2) ?

The general solution for sin(4x)=cos(x/2) is No Solution for x\in\mathbb{R}

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sin(4x)=cos(x/2)

Solution

sin (4 x) = cos (\frac{x}{2})

Solution

No Solution for x \in R

Solution steps

sin (4 x) = cos (\frac{x}{2})

Rewrite using trig identities

sin (4 x) = cos (\frac{x}{2})

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

Apply trig inverse properties

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

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

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn, 4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn, 4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn : x = \frac{π + 4 πn}{9}

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

Move

\frac{x}{2}

to the left side

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

Add

\frac{x}{2}

to both sides

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

Simplify

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

Simplify

4 x + \frac{x}{2} : \frac{9 x}{2}

4 x + \frac{x}{2}

Convert element to fraction:

4 x = \frac{4 x 2}{2}

= \frac{x}{2} + \frac{4 x \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{x + 4 x \cdot 2}{2}

x + 4 x \cdot 2 = 9 x

x + 4 x \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= x + 8 x

Add similar elements:

x + 8 x = 9 x

= 9 x

= \frac{9 x}{2}

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{9 x}{2} = \frac{π}{2} + 2 πn

\frac{9 x}{2} = \frac{π}{2} + 2 πn

\frac{9 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{9 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 9 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 9 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 9 x}{2} : 9 x

\frac{2 \cdot 9 x}{2}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{18 x}{2}

Divide the numbers:

\frac{18}{2} = 9

= 9 x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

9 x = π + 4 πn

9 x = π + 4 πn

9 x = π + 4 πn

Divide both sides by

9

9 x = π + 4 πn

Divide both sides by

9

\frac{9 x}{9} = \frac{π}{9} + \frac{4 πn}{9}

Simplify

\frac{9 x}{9} = \frac{π}{9} + \frac{4 πn}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{π}{9} + \frac{4 πn}{9} : \frac{π + 4 πn}{9}

\frac{π}{9} + \frac{4 πn}{9}

Apply rule

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

= \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn : x = \frac{π + 4 πn}{7}

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

Move

(\frac{π}{2} - \frac{x}{2})

to the left side

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

Add

(\frac{π}{2} - \frac{x}{2})

to both sides

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

Simplify

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

Simplify

4 x + \frac{π}{2} - \frac{x}{2} : \frac{π + 7 x}{2}

4 x + \frac{π}{2} - \frac{x}{2}

Combine the fractions

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

Apply rule

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

= \frac{π - x}{2}

= 4 x + \frac{- x + π}{2}

Convert element to fraction:

4 x = \frac{4 x 2}{2}

= \frac{π - x}{2} + \frac{4 x \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 - x + 4 x \cdot 2}{2}

π - x + 4 x \cdot 2 = π + 7 x

π - x + 4 x \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= π - x + 8 x

Add similar elements:

- x + 8 x = 7 x

= π + 7 x

= \frac{π + 7 x}{2}

Simplify

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

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

Add similar elements:

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

= π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{π + 7 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( π + 7 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

π + 7 x = 2 π + 4 πn

π + 7 x = 2 π + 4 πn

Move

π

to the right side

π + 7 x = 2 π + 4 πn

Subtract

π

from both sides

π + 7 x - π = 2 π + 4 πn - π

Simplify

7 x = π + 4 πn

7 x = π + 4 πn

Divide both sides by

7

7 x = π + 4 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{π}{7} + \frac{4 πn}{7}

Simplify

\frac{7 x}{7} = \frac{π}{7} + \frac{4 πn}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{π}{7} + \frac{4 πn}{7} : \frac{π + 4 πn}{7}

\frac{π}{7} + \frac{4 πn}{7}

Apply rule

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

= \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

Since the equation is undefined for:

\frac{π + 4 πn}{9}, \frac{π + 4 πn}{7}

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x)=cos(x/2) ?
The general solution for sin(4x)=cos(x/2) is No Solution for x\in\mathbb{R}