What is the general solution for sin(x+b)=cos(x) ?

The general solution for sin(x+b)=cos(x) is x=-2pin-b/2+pi/4 ,x=-(3pi)/4-2pin-b/2

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sin(x+b)=cos(x)

Solution

sin (x + b) = cos (x)

Solution

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

Solution steps

sin (x + b) = cos (x)

Subtract

cos (x)

from both sides

sin (x + b) - cos (x) = 0

Rewrite using trig identities

- cos (x) + sin (b + x)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

- (b + x) : - b - x

- (b + x)

Distribute parentheses

= - (b) - (x)

Apply minus-plus rules

+ (- a) = - a

= - b - x

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

Use the Sum to Product identity:

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

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

Simplify

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

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

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

\frac{\frac{π}{2} - b - x + x}{2}

Add similar elements:

- x + x = 0

= \frac{\frac{π}{2} - b}{2}

Join

\frac{π}{2} - b : \frac{π - 2 b}{2}

\frac{π}{2} - b

Convert element to fraction:

b = \frac{b 2}{2}

= \frac{π}{2} - \frac{b \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 - b \cdot 2}{2}

= \frac{\frac{π - 2 b}{2}}{2}

Apply the fraction rule:

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

= \frac{π - b \cdot 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π - 2 b}{4}

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

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

\frac{\frac{π}{2} - b - x - x}{2}

Add similar elements:

- x - x = - 2 x

= \frac{\frac{π}{2} - b - 2 x}{2}

Join

\frac{π}{2} - b - 2 x : \frac{π - 2 b - 4 x}{2}

\frac{π}{2} - b - 2 x

Convert element to fraction:

b = \frac{b 2}{2}, 2 x = \frac{2 x 2}{2}

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π - 2 b - 4 x}{2}

= \frac{\frac{π - 2 b - 4 x}{2}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π - 2 b - 4 x}{4}

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

sin (\frac{π - 2 b - 4 x}{4}) = 0

General solutions for

sin (\frac{π - 2 b - 4 x}{4}) = 0

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{π - 2 b - 4 x}{4} = 0 + 2 πn, \frac{π - 2 b - 4 x}{4} = π + 2 πn

\frac{π - 2 b - 4 x}{4} = 0 + 2 πn, \frac{π - 2 b - 4 x}{4} = π + 2 πn

Solve

\frac{π - 2 b - 4 x}{4} = 0 + 2 πn : x = - 2 πn - \frac{b}{2} + \frac{π}{4}

\frac{π - 2 b - 4 x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{π - 2 b - 4 x}{4} = 2 πn

Multiply both sides by

4

\frac{π - 2 b - 4 x}{4} = 2 πn

Multiply both sides by

4

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

Simplify

π - 2 b - 4 x = 8 πn

π - 2 b - 4 x = 8 πn

Move

2 b

to the right side

π - 2 b - 4 x = 8 πn

Add

2 b

to both sides

π - 2 b - 4 x + 2 b = 8 πn + 2 b

Simplify

π - 4 x = 8 πn + 2 b

π - 4 x = 8 πn + 2 b

Move

π

to the right side

π - 4 x = 8 πn + 2 b

Subtract

π

from both sides

π - 4 x - π = 8 πn + 2 b - π

Simplify

- 4 x = 8 πn + 2 b - π

- 4 x = 8 πn + 2 b - π

Divide both sides by

- 4

- 4 x = 8 πn + 2 b - π

Divide both sides by

- 4

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

Simplify

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

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{8 πn}{- 4} + \frac{2 b}{- 4} - \frac{π}{- 4} : - 2 πn - \frac{b}{2} + \frac{π}{4}

\frac{8 πn}{- 4} + \frac{2 b}{- 4} - \frac{π}{- 4}

\frac{8 πn}{- 4} = - 2 πn

\frac{8 πn}{- 4}

Apply the fraction rule:

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

= - \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2 πn

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

\frac{2 b}{- 4} = - \frac{b}{2}

\frac{2 b}{- 4}

Apply the fraction rule:

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

= - \frac{2 b}{4}

Cancel the common factor:

2

= - \frac{b}{2}

= - 2 πn - \frac{b}{2} - \frac{π}{- 4}

Refine

= - 2 πn - \frac{b}{2} + \frac{π}{4}

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

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

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

Solve

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

\frac{π - 2 b - 4 x}{4} = π + 2 πn

Multiply both sides by

4

\frac{π - 2 b - 4 x}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

π - 2 b - 4 x = 4 π + 8 πn

π - 2 b - 4 x = 4 π + 8 πn

Move

2 b

to the right side

π - 2 b - 4 x = 4 π + 8 πn

Add

2 b

to both sides

π - 2 b - 4 x + 2 b = 4 π + 8 πn + 2 b

Simplify

π - 4 x = 4 π + 8 πn + 2 b

π - 4 x = 4 π + 8 πn + 2 b

Move

π

to the right side

π - 4 x = 4 π + 8 πn + 2 b

Subtract

π

from both sides

π - 4 x - π = 4 π + 8 πn + 2 b - π

Simplify

- 4 x = 3 π + 8 πn + 2 b

- 4 x = 3 π + 8 πn + 2 b

Divide both sides by

- 4

- 4 x = 3 π + 8 πn + 2 b

Divide both sides by

- 4

\frac{- 4 x}{- 4} = \frac{3 π}{- 4} + \frac{8 πn}{- 4} + \frac{2 b}{- 4}

Simplify

\frac{- 4 x}{- 4} = \frac{3 π}{- 4} + \frac{8 πn}{- 4} + \frac{2 b}{- 4}

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{3 π}{- 4} + \frac{8 πn}{- 4} + \frac{2 b}{- 4} : - \frac{3 π}{4} - 2 πn - \frac{b}{2}

\frac{3 π}{- 4} + \frac{8 πn}{- 4} + \frac{2 b}{- 4}

Apply the fraction rule:

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

= - \frac{3 π}{4} + \frac{8 πn}{- 4} + \frac{2 b}{- 4}

\frac{8 πn}{- 4} = - 2 πn

\frac{8 πn}{- 4}

Apply the fraction rule:

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

= - \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2 πn

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

\frac{2 b}{- 4} = - \frac{b}{2}

\frac{2 b}{- 4}

Apply the fraction rule:

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

= - \frac{2 b}{4}

Cancel the common factor:

2

= - \frac{b}{2}

= - \frac{3 π}{4} - 2 πn - \frac{b}{2}

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

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+b)=cos(x) ?
The general solution for sin(x+b)=cos(x) is x=-2pin-b/2+pi/4 ,x=-(3pi)/4-2pin-b/2