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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

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

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

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

2 x = \frac{π}{2} - \frac{x}{2} + 2 πn : x = \frac{π + 4 πn}{5}

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

Move

\frac{x}{2}

to the left side

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

Add

\frac{x}{2}

to both sides

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

Simplify

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

Simplify

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

2 x + \frac{x}{2}

Convert element to fraction:

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

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

x + 2 x \cdot 2 = 5 x

x + 2 x \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= x + 4 x

Add similar elements:

x + 4 x = 5 x

= 5 x

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

\frac{5 x}{2} = \frac{π}{2} + 2 πn

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 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

5 x = π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

Divide both sides by

5

5 x = π + 4 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5}

Simplify

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{π}{5} + \frac{4 πn}{5} : \frac{π + 4 πn}{5}

\frac{π}{5} + \frac{4 πn}{5}

Apply rule

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

= \frac{π + 4 πn}{5}

x = \frac{π + 4 πn}{5}

x = \frac{π + 4 πn}{5}

x = \frac{π + 4 πn}{5}

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

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

Move

(\frac{π}{2} - \frac{x}{2})

to the left side

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

Add

(\frac{π}{2} - \frac{x}{2})

to both sides

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

Simplify

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

Simplify

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

2 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}

= 2 x + \frac{- x + π}{2}

Convert element to fraction:

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

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

π - x + 2 x \cdot 2 = π + 3 x

π - x + 2 x \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π - x + 4 x

Add similar elements:

- x + 4 x = 3 x

= π + 3 x

= \frac{π + 3 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 + 3 x}{2} = π + 2 πn

\frac{π + 3 x}{2} = π + 2 πn

\frac{π + 3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{π + 3 x}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

π + 3 x = 2 π + 4 πn

π + 3 x = 2 π + 4 πn

Move

π

to the right side

π + 3 x = 2 π + 4 πn

Subtract

π

from both sides

π + 3 x - π = 2 π + 4 πn - π

Simplify

3 x = π + 4 πn

3 x = π + 4 πn

Divide both sides by

3

3 x = π + 4 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

\frac{π}{3} + \frac{4 πn}{3}

Apply rule

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

= \frac{π + 4 πn}{3}

x = \frac{π + 4 πn}{3}

x = \frac{π + 4 πn}{3}

x = \frac{π + 4 πn}{3}

Since the equation is undefined for:

\frac{π + 4 πn}{5}, \frac{π + 4 πn}{3}

No Solution for x \in R

Popular Examples

5-7sin(x)-2cos^2(x)=0

5 - 7 sin (x) - 2 cos^{2} (x) = 0

2+cos(θ)=0

2 + cos (θ) = 0

(1+sin(x))/(sin(x))=1+cos(x)

\frac{1 + sin ( x )}{sin ( x )} = 1 + cos (x)

cos(3x)-cos(5x)=0

cos (3 x) - cos (5 x) = 0

2=2cos(x)

2 = 2 cos (x)

Frequently Asked Questions (FAQ)

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