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

The general solution for cos(x)-sin(x)=sqrt(2) is x=(7pi)/4+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(x)-sin(x)=sqrt(2)

Solution

cos (x) - sin (x) = 2

Solution

x = \frac{7 π}{4} + 2 πn

Degrees

x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) - sin (x) = 2

Subtract

2

from both sides

cos (x) - sin (x) - 2 = 0

Rewrite using trig identities

cos (x) - sin (x) - 2

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (x) - cos (\frac{π}{2} - x) - 2

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

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

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

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

\frac{x + \frac{π}{2} - x}{2}

x + \frac{π}{2} - x = \frac{π}{2}

x + \frac{π}{2} - x

Group like terms

= x - x + \frac{π}{2}

Add similar elements:

x - x = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

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

\frac{x - ( \frac{π}{2} - x )}{2}

Expand

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

x - (\frac{π}{2} - x)

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

- (\frac{π}{2} - x)

Distribute parentheses

= - (\frac{π}{2}) - (- x)

Apply minus-plus rules

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

= - \frac{π}{2} + x

= x - \frac{π}{2} + x

Simplify

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

x - \frac{π}{2} + x

Group like terms

= x + x - \frac{π}{2}

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{2}

= 2 x - \frac{π}{2}

= \frac{2 x - \frac{π}{2}}{2}

Join

2 x - \frac{π}{2} : \frac{4 x - π}{2}

2 x - \frac{π}{2}

Convert element to fraction:

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

= \frac{2 x \cdot 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 2 - π}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{2}

= \frac{\frac{4 x - π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 x - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{4}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 sin (\frac{4 x - π}{4})

= - 2 - 2 sin (\frac{4 x - π}{4})

- 2 - 2 sin (\frac{4 x - π}{4}) = 0

Move

2

to the right side

- 2 - 2 sin (\frac{4 x - π}{4}) = 0

Add

2

to both sides

- 2 - 2 sin (\frac{4 x - π}{4}) + 2 = 0 + 2

Simplify

- 2 sin (\frac{4 x - π}{4}) = 2

- 2 sin (\frac{4 x - π}{4}) = 2

Divide both sides by

- 2

- 2 sin (\frac{4 x - π}{4}) = 2

Divide both sides by

- 2

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

Simplify

sin (\frac{4 x - π}{4}) = - 1

sin (\frac{4 x - π}{4}) = - 1

General solutions for

sin (\frac{4 x - π}{4}) = - 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

\frac{4 x - π}{4} = \frac{3 π}{2} + 2 πn : x = \frac{7 π}{4} + 2 πn

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 x - π )}{4} : 4 x - π

\frac{4 ( 4 x - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 x - π

Simplify

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

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

4 x - π = 6 π + 8 πn

4 x - π = 6 π + 8 πn

4 x - π = 6 π + 8 πn

Move

π

to the right side

4 x - π = 6 π + 8 πn

Add

π

to both sides

4 x - π + π = 6 π + 8 πn + π

Simplify

4 x = 7 π + 8 πn

4 x = 7 π + 8 πn

Divide both sides by

4

4 x = 7 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{7 π}{4} + \frac{8 πn}{4}

Simplify

x = \frac{7 π}{4} + 2 πn

x = \frac{7 π}{4} + 2 πn

x = \frac{7 π}{4} + 2 πn

Popular Examples

cot(θ)=(5.8473)

cot (θ) = (5.8473)

6=2+5arctan(x/2)

6 = 2 + 5 arctan (\frac{x}{2})

tan^2(θ)-2sec(θ)=2

tan^{2} (θ) - 2 sec (θ) = 2

-sin(B)-4=3sin(B)-5

- sin (B) - 4 = 3 sin (B) - 5

cos^2(x)-7cos(x)-1=0

cos^{2} (x) - 7 cos (x) - 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)-sin(x)=sqrt(2) ?
The general solution for cos(x)-sin(x)=sqrt(2) is x=(7pi)/4+2pin