What is the general solution for sin(x-4)=cos(9x+4) ?

The general solution for sin(x-4)=cos(9x+4) is x=(4pin+pi)/(20),x=-(4pin+16+pi)/(16)

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sin(x-4)=cos(9x+4)

Solution

sin (x - 4) = cos (9 x + 4)

Solution

x = \frac{4 πn + π}{20}, x = - \frac{4 πn + 16 + π}{16}

Degrees

x = 9^{\circ} + 3 6^{\circ} n, x = - 68.54577 \dots^{\circ} - 4 5^{\circ} n

Solution steps

sin (x - 4) = cos (9 x + 4)

Rewrite using trig identities

sin (x - 4) = cos (9 x + 4)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (x - 4) = sin (\frac{π}{2} - (9 x + 4))

sin (x - 4) = sin (\frac{π}{2} - (9 x + 4))

Apply trig inverse properties

sin (x - 4) = sin (\frac{π}{2} - (9 x + 4))

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

x - 4 = \frac{π}{2} - (9 x + 4) + 2 πn, x - 4 = π - (\frac{π}{2} - (9 x + 4)) + 2 πn

x - 4 = \frac{π}{2} - (9 x + 4) + 2 πn, x - 4 = π - (\frac{π}{2} - (9 x + 4)) + 2 πn

x - 4 = \frac{π}{2} - (9 x + 4) + 2 πn : x = \frac{4 πn + π}{20}

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

Expand

\frac{π}{2} - (9 x + 4) + 2 πn : \frac{π}{2} - 9 x - 4 + 2 πn

\frac{π}{2} - (9 x + 4) + 2 πn

- (9 x + 4) : - 9 x - 4

- (9 x + 4)

Distribute parentheses

= - (9 x) - (4)

Apply minus-plus rules

+ (- a) = - a

= - 9 x - 4

= \frac{π}{2} - 9 x - 4 + 2 πn

x - 4 = \frac{π}{2} - 9 x - 4 + 2 πn

Move

4

to the right side

x - 4 = \frac{π}{2} - 9 x - 4 + 2 πn

Add

4

to both sides

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

Simplify

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

Simplify

x - 4 + 4 : x

x - 4 + 4

Add similar elements:

- 4 + 4 = 0

= x

Simplify

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

\frac{π}{2} - 9 x - 4 + 2 πn + 4

Group like terms

= - 9 x + 2 πn + \frac{π}{2} - 4 + 4

- 4 + 4 = 0

= - 9 x + 2 πn + \frac{π}{2}

x = - 9 x + 2 πn + \frac{π}{2}

x = - 9 x + 2 πn + \frac{π}{2}

x = - 9 x + 2 πn + \frac{π}{2}

Move

9 x

to the left side

x = - 9 x + 2 πn + \frac{π}{2}

Add

9 x

to both sides

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

Simplify

10 x = 2 πn + \frac{π}{2}

10 x = 2 πn + \frac{π}{2}

Divide both sides by

10

10 x = 2 πn + \frac{π}{2}

Divide both sides by

10

\frac{10 x}{10} = \frac{2 πn}{10} + \frac{\frac{π}{2}}{10}

Simplify

\frac{10 x}{10} = \frac{2 πn}{10} + \frac{\frac{π}{2}}{10}

Simplify

\frac{10 x}{10} : x

\frac{10 x}{10}

Divide the numbers:

\frac{10}{10} = 1

= x

Simplify

\frac{2 πn}{10} + \frac{\frac{π}{2}}{10} : \frac{4 πn + π}{20}

\frac{2 πn}{10} + \frac{\frac{π}{2}}{10}

Apply rule

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

= \frac{2 πn + \frac{π}{2}}{10}

Join

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

2 πn + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{2 πn \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}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 πn + π}{2}

= \frac{\frac{4 πn + π}{2}}{10}

Apply the fraction rule:

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

= \frac{4 πn + π}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{4 πn + π}{20}

x = \frac{4 πn + π}{20}

x = \frac{4 πn + π}{20}

x = \frac{4 πn + π}{20}

x - 4 = π - (\frac{π}{2} - (9 x + 4)) + 2 πn : x = - \frac{4 πn + 16 + π}{16}

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

Expand

π - (\frac{π}{2} - (9 x + 4)) + 2 πn : π - \frac{π}{2} + 9 x + 4 + 2 πn

π - (\frac{π}{2} - (9 x + 4)) + 2 πn

- (9 x + 4) : - 9 x - 4

- (9 x + 4)

Distribute parentheses

= - (9 x) - (4)

Apply minus-plus rules

+ (- a) = - a

= - 9 x - 4

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

- (\frac{π}{2} - 9 x - 4) : - \frac{π}{2} + 9 x + 4

- (\frac{π}{2} - 9 x - 4)

Distribute parentheses

= - (\frac{π}{2}) - (- 9 x) - (- 4)

Apply minus-plus rules

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

= - \frac{π}{2} + 9 x + 4

= π - \frac{π}{2} + 9 x + 4 + 2 πn

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

Move

4

to the right side

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

Add

4

to both sides

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

Simplify

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

Simplify

x - 4 + 4 : x

x - 4 + 4

Add similar elements:

- 4 + 4 = 0

= x

Simplify

π - \frac{π}{2} + 9 x + 4 + 2 πn + 4 : 9 x + 2 πn + 8 + π - \frac{π}{2}

π - \frac{π}{2} + 9 x + 4 + 2 πn + 4

Group like terms

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

Add the numbers:

4 + 4 = 8

= 9 x + 2 πn + 8 + π - \frac{π}{2}

x = 9 x + 2 πn + 8 + π - \frac{π}{2}

x = 9 x + 2 πn + 8 + π - \frac{π}{2}

x = 9 x + 2 πn + 8 + π - \frac{π}{2}

Move

9 x

to the left side

x = 9 x + 2 πn + 8 + π - \frac{π}{2}

Subtract

9 x

from both sides

x - 9 x = 9 x + 2 πn + 8 + π - \frac{π}{2} - 9 x

Simplify

- 8 x = 2 πn + 8 + π - \frac{π}{2}

- 8 x = 2 πn + 8 + π - \frac{π}{2}

Divide both sides by

- 8

- 8 x = 2 πn + 8 + π - \frac{π}{2}

Divide both sides by

- 8

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

Simplify

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

Simplify

\frac{- 8 x}{- 8} : x

\frac{- 8 x}{- 8}

Apply the fraction rule:

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

= \frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{2 πn}{- 8} + \frac{8}{- 8} + \frac{π}{- 8} - \frac{\frac{π}{2}}{- 8} : - \frac{4 πn + 16 + π}{16}

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

Apply rule

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

= \frac{2 πn + 8 + π - \frac{π}{2}}{- 8}

Apply the fraction rule:

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

= - \frac{2 πn + 8 + π - \frac{π}{2}}{8}

Join

2 πn + 8 + π - \frac{π}{2} : \frac{4 πn + 16 + π}{2}

2 πn + 8 + π - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 8 = \frac{8 \cdot 2}{2}, π = \frac{π 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{8 \cdot 2}{2} + \frac{π 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 + 8 \cdot 2 + π 2 - π}{2}

2 πn \cdot 2 + 8 \cdot 2 + π 2 - π = 4 πn + 16 + π

2 πn \cdot 2 + 8 \cdot 2 + π 2 - π

Add similar elements:

2 π - π = π

= 2 \cdot 2 πn + 8 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 8 \cdot 2 + π

Multiply the numbers:

8 \cdot 2 = 16

= 4 πn + 16 + π

= \frac{4 πn + 16 + π}{2}

= - \frac{\frac{4 πn + π + 16}{2}}{8}

Simplify

\frac{\frac{4 πn + 16 + π}{2}}{8} : \frac{4 πn + 16 + π}{16}

\frac{\frac{4 πn + 16 + π}{2}}{8}

Apply the fraction rule:

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

= \frac{4 πn + 16 + π}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{4 πn + 16 + π}{16}

= - \frac{4 πn + π + 16}{16}

= - \frac{4 πn + 16 + π}{16}

x = - \frac{4 πn + 16 + π}{16}

x = - \frac{4 πn + 16 + π}{16}

x = - \frac{4 πn + 16 + π}{16}

x = \frac{4 πn + π}{20}, x = - \frac{4 πn + 16 + π}{16}

x = \frac{4 πn + π}{20}, x = - \frac{4 πn + 16 + π}{16}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-4)=cos(9x+4) ?
The general solution for sin(x-4)=cos(9x+4) is x=(4pin+pi)/(20),x=-(4pin+16+pi)/(16)