What is the general solution for sin(5x+8)=cos(9x-16) ?

The general solution for sin(5x+8)=cos(9x-16) is x=(4pin+16+pi}{28},x=-\frac{pi+4pin-48)/8

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sin(5x+8)=cos(9x-16)

Solution

sin (5 x + 8) = cos (9 x - 16)

Solution

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

Degrees

x = 39.16901 \dots^{\circ} + 25.71428 \dots^{\circ} n, x = 321.27467 \dots^{\circ} - 9 0^{\circ} n

Solution steps

sin (5 x + 8) = cos (9 x - 16)

Rewrite using trig identities

sin (5 x + 8) = cos (9 x - 16)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

Apply trig inverse properties

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

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

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn, 5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn, 5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn : x = \frac{4 πn + 16 + π}{28}

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn

Expand

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

\frac{π}{2} - (9 x - 16) + 2 πn

- (9 x - 16) : - 9 x + 16

- (9 x - 16)

Distribute parentheses

= - (9 x) - (- 16)

Apply minus-plus rules

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

= - 9 x + 16

= \frac{π}{2} - 9 x + 16 + 2 πn

5 x + 8 = \frac{π}{2} - 9 x + 16 + 2 πn

Move

8

to the right side

5 x + 8 = \frac{π}{2} - 9 x + 16 + 2 πn

Subtract

8

from both sides

5 x + 8 - 8 = \frac{π}{2} - 9 x + 16 + 2 πn - 8

Simplify

5 x + 8 - 8 = \frac{π}{2} - 9 x + 16 + 2 πn - 8

Simplify

5 x + 8 - 8 : 5 x

5 x + 8 - 8

Add similar elements:

8 - 8 = 0

= 5 x

Simplify

\frac{π}{2} - 9 x + 16 + 2 πn - 8 : - 9 x + 2 πn + 8 + \frac{π}{2}

\frac{π}{2} - 9 x + 16 + 2 πn - 8

Group like terms

= - 9 x + 2 πn + \frac{π}{2} + 16 - 8

Add/Subtract the numbers:

16 - 8 = 8

= - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

Move

9 x

to the left side

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

Add

9 x

to both sides

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

Simplify

14 x = 2 πn + 8 + \frac{π}{2}

14 x = 2 πn + 8 + \frac{π}{2}

Divide both sides by

14

14 x = 2 πn + 8 + \frac{π}{2}

Divide both sides by

14

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{8}{14} + \frac{\frac{π}{2}}{14}

Simplify

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{8}{14} + \frac{\frac{π}{2}}{14}

Simplify

\frac{14 x}{14} : x

\frac{14 x}{14}

Divide the numbers:

\frac{14}{14} = 1

= x

Simplify

\frac{2 πn}{14} + \frac{8}{14} + \frac{\frac{π}{2}}{14} : \frac{4 πn + 16 + π}{28}

\frac{2 πn}{14} + \frac{8}{14} + \frac{\frac{π}{2}}{14}

Apply rule

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

= \frac{2 πn + 8 + \frac{π}{2}}{14}

Join

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

2 πn + 8 + \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + 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 + 16 + π}{2}}{14}

Apply the fraction rule:

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

= \frac{4 πn + 16 + π}{2 \cdot 14}

Multiply the numbers:

2 \cdot 14 = 28

= \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn : x = - \frac{π + 4 πn - 48}{8}

5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

Expand

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

π - (\frac{π}{2} - (9 x - 16)) + 2 πn

- (9 x - 16) : - 9 x + 16

- (9 x - 16)

Distribute parentheses

= - (9 x) - (- 16)

Apply minus-plus rules

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

= - 9 x + 16

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

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

- (\frac{π}{2} - 9 x + 16)

Distribute parentheses

= - (\frac{π}{2}) - (- 9 x) - (16)

Apply minus-plus rules

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

= - \frac{π}{2} + 9 x - 16

= π - \frac{π}{2} + 9 x - 16 + 2 πn

5 x + 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn

Move

8

to the right side

5 x + 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn

Subtract

8

from both sides

5 x + 8 - 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

Simplify

5 x + 8 - 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

Simplify

5 x + 8 - 8 : 5 x

5 x + 8 - 8

Add similar elements:

8 - 8 = 0

= 5 x

Simplify

π - \frac{π}{2} + 9 x - 16 + 2 πn - 8 : 9 x + 2 πn + π - 24 - \frac{π}{2}

π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

Group like terms

= 9 x + π + 2 πn - \frac{π}{2} - 16 - 8

Subtract the numbers:

- 16 - 8 = - 24

= 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

Move

9 x

to the left side

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

Subtract

9 x

from both sides

5 x - 9 x = 9 x + 2 πn + π - 24 - \frac{π}{2} - 9 x

Simplify

- 4 x = 2 πn + π - 24 - \frac{π}{2}

- 4 x = 2 πn + π - 24 - \frac{π}{2}

Divide both sides by

- 4

- 4 x = 2 πn + π - 24 - \frac{π}{2}

Divide both sides by

- 4

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4}

Simplify

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 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{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4} : - \frac{π + 4 πn - 48}{8}

\frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4}

Apply rule

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

= \frac{2 πn + π - 24 - \frac{π}{2}}{- 4}

Apply the fraction rule:

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

= - \frac{2 πn + π - 24 - \frac{π}{2}}{4}

Join

2 πn + π - 24 - \frac{π}{2} : \frac{π + 4 πn - 48}{2}

2 πn + π - 24 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 24 = \frac{24 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{24 \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 - 24 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 24 \cdot 2 - π = π + 4 πn - 48

2 πn \cdot 2 + π 2 - 24 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 24 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 24 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 24 \cdot 2

Multiply the numbers:

24 \cdot 2 = 48

= π + 4 πn - 48

= \frac{π + 4 πn - 48}{2}

= - \frac{\frac{π + 4 πn - 48}{2}}{4}

Simplify

\frac{\frac{π + 4 πn - 48}{2}}{4} : \frac{π + 4 πn - 48}{8}

\frac{\frac{π + 4 πn - 48}{2}}{4}

Apply the fraction rule:

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

= \frac{π + 4 πn - 48}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 4 πn - 48}{8}

= - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x+8)=cos(9x-16) ?
The general solution for sin(5x+8)=cos(9x-16) is x=(4pin+16+pi}{28},x=-\frac{pi+4pin-48)/8