What is the general solution for sin(4x-3)=cos(5x-6) ?

The general solution for sin(4x-3)=cos(5x-6) is x=(4pin+18+pi}{18},x=-\frac{pi+4pin-6)/2

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sin(4x-3)=cos(5x-6)

Solution

sin (4 x - 3) = cos (5 x - 6)

Solution

x = \frac{4 πn + 18 + π}{18}, x = - \frac{π + 4 πn - 6}{2}

Degrees

x = 67.29577 \dots^{\circ} + 4 0^{\circ} n, x = 81.88733 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (4 x - 3) = cos (5 x - 6)

Rewrite using trig identities

sin (4 x - 3) = cos (5 x - 6)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (4 x - 3) = sin (\frac{π}{2} - (5 x - 6))

sin (4 x - 3) = sin (\frac{π}{2} - (5 x - 6))

Apply trig inverse properties

sin (4 x - 3) = sin (\frac{π}{2} - (5 x - 6))

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

4 x - 3 = \frac{π}{2} - (5 x - 6) + 2 πn, 4 x - 3 = π - (\frac{π}{2} - (5 x - 6)) + 2 πn

4 x - 3 = \frac{π}{2} - (5 x - 6) + 2 πn, 4 x - 3 = π - (\frac{π}{2} - (5 x - 6)) + 2 πn

4 x - 3 = \frac{π}{2} - (5 x - 6) + 2 πn : x = \frac{4 πn + 18 + π}{18}

4 x - 3 = \frac{π}{2} - (5 x - 6) + 2 πn

Expand

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

\frac{π}{2} - (5 x - 6) + 2 πn

- (5 x - 6) : - 5 x + 6

- (5 x - 6)

Distribute parentheses

= - (5 x) - (- 6)

Apply minus-plus rules

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

= - 5 x + 6

= \frac{π}{2} - 5 x + 6 + 2 πn

4 x - 3 = \frac{π}{2} - 5 x + 6 + 2 πn

Move

3

to the right side

4 x - 3 = \frac{π}{2} - 5 x + 6 + 2 πn

Add

3

to both sides

4 x - 3 + 3 = \frac{π}{2} - 5 x + 6 + 2 πn + 3

Simplify

4 x - 3 + 3 = \frac{π}{2} - 5 x + 6 + 2 πn + 3

Simplify

4 x - 3 + 3 : 4 x

4 x - 3 + 3

Add similar elements:

- 3 + 3 = 0

= 4 x

Simplify

\frac{π}{2} - 5 x + 6 + 2 πn + 3 : - 5 x + 2 πn + 9 + \frac{π}{2}

\frac{π}{2} - 5 x + 6 + 2 πn + 3

Group like terms

= - 5 x + 2 πn + \frac{π}{2} + 6 + 3

Add the numbers:

6 + 3 = 9

= - 5 x + 2 πn + 9 + \frac{π}{2}

4 x = - 5 x + 2 πn + 9 + \frac{π}{2}

4 x = - 5 x + 2 πn + 9 + \frac{π}{2}

4 x = - 5 x + 2 πn + 9 + \frac{π}{2}

Move

5 x

to the left side

4 x = - 5 x + 2 πn + 9 + \frac{π}{2}

Add

5 x

to both sides

4 x + 5 x = - 5 x + 2 πn + 9 + \frac{π}{2} + 5 x

Simplify

9 x = 2 πn + 9 + \frac{π}{2}

9 x = 2 πn + 9 + \frac{π}{2}

Divide both sides by

9

9 x = 2 πn + 9 + \frac{π}{2}

Divide both sides by

9

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

Simplify

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

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{2 πn}{9} + \frac{9}{9} + \frac{\frac{π}{2}}{9} : \frac{4 πn + 18 + π}{18}

\frac{2 πn}{9} + \frac{9}{9} + \frac{\frac{π}{2}}{9}

Apply rule

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

= \frac{2 πn + 9 + \frac{π}{2}}{9}

Join

2 πn + 9 + \frac{π}{2} : \frac{4 πn + 18 + π}{2}

2 πn + 9 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 9 = \frac{9 \cdot 2}{2}

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

2 πn \cdot 2 + 9 \cdot 2 + π = 4 πn + 18 + π

2 πn \cdot 2 + 9 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 9 \cdot 2 + π

Multiply the numbers:

9 \cdot 2 = 18

= 4 πn + 18 + π

= \frac{4 πn + 18 + π}{2}

= \frac{\frac{4 πn + 18 + π}{2}}{9}

Apply the fraction rule:

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

= \frac{4 πn + 18 + π}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{4 πn + 18 + π}{18}

x = \frac{4 πn + 18 + π}{18}

x = \frac{4 πn + 18 + π}{18}

x = \frac{4 πn + 18 + π}{18}

4 x - 3 = π - (\frac{π}{2} - (5 x - 6)) + 2 πn : x = - \frac{π + 4 πn - 6}{2}

4 x - 3 = π - (\frac{π}{2} - (5 x - 6)) + 2 πn

Expand

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

π - (\frac{π}{2} - (5 x - 6)) + 2 πn

- (5 x - 6) : - 5 x + 6

- (5 x - 6)

Distribute parentheses

= - (5 x) - (- 6)

Apply minus-plus rules

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

= - 5 x + 6

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

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

- (\frac{π}{2} - 5 x + 6)

Distribute parentheses

= - (\frac{π}{2}) - (- 5 x) - (6)

Apply minus-plus rules

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

= - \frac{π}{2} + 5 x - 6

= π - \frac{π}{2} + 5 x - 6 + 2 πn

4 x - 3 = π - \frac{π}{2} + 5 x - 6 + 2 πn

Move

3

to the right side

4 x - 3 = π - \frac{π}{2} + 5 x - 6 + 2 πn

Add

3

to both sides

4 x - 3 + 3 = π - \frac{π}{2} + 5 x - 6 + 2 πn + 3

Simplify

4 x - 3 + 3 = π - \frac{π}{2} + 5 x - 6 + 2 πn + 3

Simplify

4 x - 3 + 3 : 4 x

4 x - 3 + 3

Add similar elements:

- 3 + 3 = 0

= 4 x

Simplify

π - \frac{π}{2} + 5 x - 6 + 2 πn + 3 : 5 x + 2 πn + π - 3 - \frac{π}{2}

π - \frac{π}{2} + 5 x - 6 + 2 πn + 3

Group like terms

= 5 x + π + 2 πn - \frac{π}{2} - 6 + 3

Add/Subtract the numbers:

- 6 + 3 = - 3

= 5 x + 2 πn + π - 3 - \frac{π}{2}

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

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

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

Move

5 x

to the left side

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

Subtract

5 x

from both sides

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

Simplify

- x = 2 πn + π - 3 - \frac{π}{2}

- x = 2 πn + π - 3 - \frac{π}{2}

Divide both sides by

- 1

- x = 2 πn + π - 3 - \frac{π}{2}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{3}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{3}{- 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{3}{- 1} - \frac{\frac{π}{2}}{- 1} : - \frac{π + 4 πn - 6}{2}

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{3}{- 1} - \frac{\frac{π}{2}}{- 1}

Apply rule

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

= \frac{2 πn + π - 3 - \frac{π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn + π - 3 - \frac{π}{2}}{1}

Join

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

2 πn + π - 3 - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + π 2 - 3 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 3 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 3 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= π + 4 πn - 6

= \frac{π + 4 πn - 6}{2}

= - \frac{\frac{π + 4 πn - 6}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn - 6}{2}

x = - \frac{π + 4 πn - 6}{2}

x = - \frac{π + 4 πn - 6}{2}

x = - \frac{π + 4 πn - 6}{2}

x = \frac{4 πn + 18 + π}{18}, x = - \frac{π + 4 πn - 6}{2}

x = \frac{4 πn + 18 + π}{18}, x = - \frac{π + 4 πn - 6}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x-3)=cos(5x-6) ?
The general solution for sin(4x-3)=cos(5x-6) is x=(4pin+18+pi}{18},x=-\frac{pi+4pin-6)/2