What is the general solution for sin(x+5)=cos(2x-2) ?

The general solution for sin(x+5)=cos(2x-2) is x=(4pin+pi-6)/6 ,x=-(pi+4pin-14)/2

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sin(x+5)=cos(2x-2)

Solution

sin (x + 5) = cos (2 x - 2)

Solution

x = \frac{4 πn + π - 6}{6}, x = - \frac{π + 4 πn - 14}{2}

Degrees

x = - 27.29577 \dots^{\circ} + 12 0^{\circ} n, x = 311.07045 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (x + 5) = cos (2 x - 2)

Rewrite using trig identities

sin (x + 5) = cos (2 x - 2)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (x + 5) = sin (\frac{π}{2} - (2 x - 2))

sin (x + 5) = sin (\frac{π}{2} - (2 x - 2))

Apply trig inverse properties

sin (x + 5) = sin (\frac{π}{2} - (2 x - 2))

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

x + 5 = \frac{π}{2} - (2 x - 2) + 2 πn, x + 5 = π - (\frac{π}{2} - (2 x - 2)) + 2 πn

x + 5 = \frac{π}{2} - (2 x - 2) + 2 πn, x + 5 = π - (\frac{π}{2} - (2 x - 2)) + 2 πn

x + 5 = \frac{π}{2} - (2 x - 2) + 2 πn : x = \frac{4 πn + π - 6}{6}

x + 5 = \frac{π}{2} - (2 x - 2) + 2 πn

Expand

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

\frac{π}{2} - (2 x - 2) + 2 πn

- (2 x - 2) : - 2 x + 2

- (2 x - 2)

Distribute parentheses

= - (2 x) - (- 2)

Apply minus-plus rules

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

= - 2 x + 2

= \frac{π}{2} - 2 x + 2 + 2 πn

x + 5 = \frac{π}{2} - 2 x + 2 + 2 πn

Move

5

to the right side

x + 5 = \frac{π}{2} - 2 x + 2 + 2 πn

Subtract

5

from both sides

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

Simplify

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

Simplify

x + 5 - 5 : x

x + 5 - 5

Add similar elements:

5 - 5 = 0

= x

Simplify

\frac{π}{2} - 2 x + 2 + 2 πn - 5 : - 2 x + 2 πn + \frac{π}{2} - 3

\frac{π}{2} - 2 x + 2 + 2 πn - 5

Group like terms

= - 2 x + 2 πn + \frac{π}{2} + 2 - 5

Add/Subtract the numbers:

2 - 5 = - 3

= - 2 x + 2 πn + \frac{π}{2} - 3

x = - 2 x + 2 πn + \frac{π}{2} - 3

x = - 2 x + 2 πn + \frac{π}{2} - 3

x = - 2 x + 2 πn + \frac{π}{2} - 3

Move

2 x

to the left side

x = - 2 x + 2 πn + \frac{π}{2} - 3

Add

2 x

to both sides

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

Simplify

3 x = 2 πn + \frac{π}{2} - 3

3 x = 2 πn + \frac{π}{2} - 3

Divide both sides by

3

3 x = 2 πn + \frac{π}{2} - 3

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

\frac{2 πn}{3} + \frac{\frac{π}{2}}{3} - \frac{3}{3}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 3}{3}

Join

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

2 πn + \frac{π}{2} - 3

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 3 = \frac{3 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{3 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 πn \cdot 2 + π - 3 \cdot 2}{2}

2 πn \cdot 2 + π - 3 \cdot 2 = 4 πn + π - 6

2 πn \cdot 2 + π - 3 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 4 πn + π - 6

= \frac{4 πn + π - 6}{2}

= \frac{\frac{4 πn + π - 6}{2}}{3}

Apply the fraction rule:

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

= \frac{4 πn + π - 6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4 πn + π - 6}{6}

x = \frac{4 πn + π - 6}{6}

x = \frac{4 πn + π - 6}{6}

x = \frac{4 πn + π - 6}{6}

x + 5 = π - (\frac{π}{2} - (2 x - 2)) + 2 πn : x = - \frac{π + 4 πn - 14}{2}

x + 5 = π - (\frac{π}{2} - (2 x - 2)) + 2 πn

Expand

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

π - (\frac{π}{2} - (2 x - 2)) + 2 πn

- (2 x - 2) : - 2 x + 2

- (2 x - 2)

Distribute parentheses

= - (2 x) - (- 2)

Apply minus-plus rules

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

= - 2 x + 2

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

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

- (\frac{π}{2} - 2 x + 2)

Distribute parentheses

= - (\frac{π}{2}) - (- 2 x) - (2)

Apply minus-plus rules

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

= - \frac{π}{2} + 2 x - 2

= π - \frac{π}{2} + 2 x - 2 + 2 πn

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

Move

5

to the right side

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

Subtract

5

from both sides

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

Simplify

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

Simplify

x + 5 - 5 : x

x + 5 - 5

Add similar elements:

5 - 5 = 0

= x

Simplify

π - \frac{π}{2} + 2 x - 2 + 2 πn - 5 : 2 x + 2 πn + π - 7 - \frac{π}{2}

π - \frac{π}{2} + 2 x - 2 + 2 πn - 5

Group like terms

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

Subtract the numbers:

- 2 - 5 = - 7

= 2 x + 2 πn + π - 7 - \frac{π}{2}

x = 2 x + 2 πn + π - 7 - \frac{π}{2}

x = 2 x + 2 πn + π - 7 - \frac{π}{2}

x = 2 x + 2 πn + π - 7 - \frac{π}{2}

Move

2 x

to the left side

x = 2 x + 2 πn + π - 7 - \frac{π}{2}

Subtract

2 x

from both sides

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

Simplify

- x = 2 πn + π - 7 - \frac{π}{2}

- x = 2 πn + π - 7 - \frac{π}{2}

Divide both sides by

- 1

- x = 2 πn + π - 7 - \frac{π}{2}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{7}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{7}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{7}{- 1} - \frac{\frac{π}{2}}{- 1} : - \frac{π + 4 πn - 14}{2}

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{7}{- 1} - \frac{\frac{π}{2}}{- 1}

Apply rule

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

= \frac{2 πn + π - 7 - \frac{π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn + π - 7 - \frac{π}{2}}{1}

Join

2 πn + π - 7 - \frac{π}{2} : \frac{π + 4 πn - 14}{2}

2 πn + π - 7 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 7 = \frac{7 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{7 \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 πn \cdot 2 + π 2 - 7 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 7 \cdot 2 - π = π + 4 πn - 14

2 πn \cdot 2 + π 2 - 7 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 7 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 7 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= π + 4 πn - 14

= \frac{π + 4 πn - 14}{2}

= - \frac{\frac{π + 4 πn - 14}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn - 14}{2}

x = - \frac{π + 4 πn - 14}{2}

x = - \frac{π + 4 πn - 14}{2}

x = - \frac{π + 4 πn - 14}{2}

x = \frac{4 πn + π - 6}{6}, x = - \frac{π + 4 πn - 14}{2}

x = \frac{4 πn + π - 6}{6}, x = - \frac{π + 4 πn - 14}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+5)=cos(2x-2) ?
The general solution for sin(x+5)=cos(2x-2) is x=(4pin+pi-6)/6 ,x=-(pi+4pin-14)/2