What is the general solution for cos(9x+15)=sin(7x-5) ?

The general solution for cos(9x+15)=sin(7x-5) is x=(4pin+pi-20}{32},x=-\frac{4pin+40+pi)/4

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cos(9x+15)=sin(7x-5)

Solution

cos (9 x + 15) = sin (7 x - 5)

Solution

x = \frac{4 πn + π - 20}{32}, x = - \frac{4 πn + 40 + π}{4}

Degrees

x = - 30.18486 \dots^{\circ} + 22. 5^{\circ} n, x = - 617.95779 \dots^{\circ} - 18 0^{\circ} n

Solution steps

cos (9 x + 15) = sin (7 x - 5)

Rewrite using trig identities

cos (9 x + 15) = sin (7 x - 5)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (9 x + 15) = sin (\frac{π}{2} - (9 x + 15))

cos (9 x + 15) = sin (\frac{π}{2} - (9 x + 15))

Apply trig inverse properties

cos (9 x + 15) = sin (\frac{π}{2} - (9 x + 15))

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

7 x - 5 = \frac{π}{2} - (9 x + 15) + 2 πn, 7 x - 5 = π - (\frac{π}{2} - (9 x + 15)) + 2 πn

7 x - 5 = \frac{π}{2} - (9 x + 15) + 2 πn, 7 x - 5 = π - (\frac{π}{2} - (9 x + 15)) + 2 πn

7 x - 5 = \frac{π}{2} - (9 x + 15) + 2 πn : x = \frac{4 πn + π - 20}{32}

7 x - 5 = \frac{π}{2} - (9 x + 15) + 2 πn

Expand

\frac{π}{2} - (9 x + 15) + 2 πn : \frac{π}{2} - 9 x - 15 + 2 πn

\frac{π}{2} - (9 x + 15) + 2 πn

- (9 x + 15) : - 9 x - 15

- (9 x + 15)

Distribute parentheses

= - (9 x) - (15)

Apply minus-plus rules

+ (- a) = - a

= - 9 x - 15

= \frac{π}{2} - 9 x - 15 + 2 πn

7 x - 5 = \frac{π}{2} - 9 x - 15 + 2 πn

Move

5

to the right side

7 x - 5 = \frac{π}{2} - 9 x - 15 + 2 πn

Add

5

to both sides

7 x - 5 + 5 = \frac{π}{2} - 9 x - 15 + 2 πn + 5

Simplify

7 x - 5 + 5 = \frac{π}{2} - 9 x - 15 + 2 πn + 5

Simplify

7 x - 5 + 5 : 7 x

7 x - 5 + 5

Add similar elements:

- 5 + 5 = 0

= 7 x

Simplify

\frac{π}{2} - 9 x - 15 + 2 πn + 5 : - 9 x + 2 πn + \frac{π}{2} - 10

\frac{π}{2} - 9 x - 15 + 2 πn + 5

Group like terms

= - 9 x + 2 πn + \frac{π}{2} - 15 + 5

Add/Subtract the numbers:

- 15 + 5 = - 10

= - 9 x + 2 πn + \frac{π}{2} - 10

7 x = - 9 x + 2 πn + \frac{π}{2} - 10

7 x = - 9 x + 2 πn + \frac{π}{2} - 10

7 x = - 9 x + 2 πn + \frac{π}{2} - 10

Move

9 x

to the left side

7 x = - 9 x + 2 πn + \frac{π}{2} - 10

Add

9 x

to both sides

7 x + 9 x = - 9 x + 2 πn + \frac{π}{2} - 10 + 9 x

Simplify

16 x = 2 πn + \frac{π}{2} - 10

16 x = 2 πn + \frac{π}{2} - 10

Divide both sides by

16

16 x = 2 πn + \frac{π}{2} - 10

Divide both sides by

16

\frac{16 x}{16} = \frac{2 πn}{16} + \frac{\frac{π}{2}}{16} - \frac{10}{16}

Simplify

\frac{16 x}{16} = \frac{2 πn}{16} + \frac{\frac{π}{2}}{16} - \frac{10}{16}

Simplify

\frac{16 x}{16} : x

\frac{16 x}{16}

Divide the numbers:

\frac{16}{16} = 1

= x

Simplify

\frac{2 πn}{16} + \frac{\frac{π}{2}}{16} - \frac{10}{16} : \frac{4 πn + π - 20}{32}

\frac{2 πn}{16} + \frac{\frac{π}{2}}{16} - \frac{10}{16}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 10}{16}

Join

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

2 πn + \frac{π}{2} - 10

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 10 = \frac{10 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{10 \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 - 10 \cdot 2}{2}

2 πn \cdot 2 + π - 10 \cdot 2 = 4 πn + π - 20

2 πn \cdot 2 + π - 10 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 10 \cdot 2

Multiply the numbers:

10 \cdot 2 = 20

= 4 πn + π - 20

= \frac{4 πn + π - 20}{2}

= \frac{\frac{4 πn + π - 20}{2}}{16}

Apply the fraction rule:

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

= \frac{4 πn + π - 20}{2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{4 πn + π - 20}{32}

x = \frac{4 πn + π - 20}{32}

x = \frac{4 πn + π - 20}{32}

x = \frac{4 πn + π - 20}{32}

7 x - 5 = π - (\frac{π}{2} - (9 x + 15)) + 2 πn : x = - \frac{4 πn + 40 + π}{4}

7 x - 5 = π - (\frac{π}{2} - (9 x + 15)) + 2 πn

Expand

π - (\frac{π}{2} - (9 x + 15)) + 2 πn : π - \frac{π}{2} + 9 x + 15 + 2 πn

π - (\frac{π}{2} - (9 x + 15)) + 2 πn

- (9 x + 15) : - 9 x - 15

- (9 x + 15)

Distribute parentheses

= - (9 x) - (15)

Apply minus-plus rules

+ (- a) = - a

= - 9 x - 15

= π - (- 9 x + \frac{π}{2} - 15) + 2 πn

- (\frac{π}{2} - 9 x - 15) : - \frac{π}{2} + 9 x + 15

- (\frac{π}{2} - 9 x - 15)

Distribute parentheses

= - (\frac{π}{2}) - (- 9 x) - (- 15)

Apply minus-plus rules

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

= - \frac{π}{2} + 9 x + 15

= π - \frac{π}{2} + 9 x + 15 + 2 πn

7 x - 5 = π - \frac{π}{2} + 9 x + 15 + 2 πn

Move

5

to the right side

7 x - 5 = π - \frac{π}{2} + 9 x + 15 + 2 πn

Add

5

to both sides

7 x - 5 + 5 = π - \frac{π}{2} + 9 x + 15 + 2 πn + 5

Simplify

7 x - 5 + 5 = π - \frac{π}{2} + 9 x + 15 + 2 πn + 5

Simplify

7 x - 5 + 5 : 7 x

7 x - 5 + 5

Add similar elements:

- 5 + 5 = 0

= 7 x

Simplify

π - \frac{π}{2} + 9 x + 15 + 2 πn + 5 : 9 x + 2 πn + 20 + π - \frac{π}{2}

π - \frac{π}{2} + 9 x + 15 + 2 πn + 5

Group like terms

= 9 x + π + 2 πn - \frac{π}{2} + 15 + 5

Add the numbers:

15 + 5 = 20

= 9 x + 2 πn + 20 + π - \frac{π}{2}

7 x = 9 x + 2 πn + 20 + π - \frac{π}{2}

7 x = 9 x + 2 πn + 20 + π - \frac{π}{2}

7 x = 9 x + 2 πn + 20 + π - \frac{π}{2}

Move

9 x

to the left side

7 x = 9 x + 2 πn + 20 + π - \frac{π}{2}

Subtract

9 x

from both sides

7 x - 9 x = 9 x + 2 πn + 20 + π - \frac{π}{2} - 9 x

Simplify

- 2 x = 2 πn + 20 + π - \frac{π}{2}

- 2 x = 2 πn + 20 + π - \frac{π}{2}

Divide both sides by

- 2

- 2 x = 2 πn + 20 + π - \frac{π}{2}

Divide both sides by

- 2

\frac{- 2 x}{- 2} = \frac{2 πn}{- 2} + \frac{20}{- 2} + \frac{π}{- 2} - \frac{\frac{π}{2}}{- 2}

Simplify

\frac{- 2 x}{- 2} = \frac{2 πn}{- 2} + \frac{20}{- 2} + \frac{π}{- 2} - \frac{\frac{π}{2}}{- 2}

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{- 2} + \frac{20}{- 2} + \frac{π}{- 2} - \frac{\frac{π}{2}}{- 2} : - \frac{4 πn + 40 + π}{4}

\frac{2 πn}{- 2} + \frac{20}{- 2} + \frac{π}{- 2} - \frac{\frac{π}{2}}{- 2}

Apply rule

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

= \frac{2 πn + 20 + π - \frac{π}{2}}{- 2}

Apply the fraction rule:

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

= - \frac{2 πn + 20 + π - \frac{π}{2}}{2}

Join

2 πn + 20 + π - \frac{π}{2} : \frac{4 πn + 40 + π}{2}

2 πn + 20 + π - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 20 = \frac{20 \cdot 2}{2}, π = \frac{π 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{20 \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 + 20 \cdot 2 + π 2 - π}{2}

2 πn \cdot 2 + 20 \cdot 2 + π 2 - π = 4 πn + 40 + π

2 πn \cdot 2 + 20 \cdot 2 + π 2 - π

Add similar elements:

2 π - π = π

= 2 \cdot 2 πn + 20 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 20 \cdot 2 + π

Multiply the numbers:

20 \cdot 2 = 40

= 4 πn + 40 + π

= \frac{4 πn + 40 + π}{2}

= - \frac{\frac{4 πn + π + 40}{2}}{2}

Simplify

\frac{\frac{4 πn + 40 + π}{2}}{2} : \frac{4 πn + 40 + π}{4}

\frac{\frac{4 πn + 40 + π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 πn + 40 + π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 πn + 40 + π}{4}

= - \frac{4 πn + π + 40}{4}

= - \frac{4 πn + 40 + π}{4}

x = - \frac{4 πn + 40 + π}{4}

x = - \frac{4 πn + 40 + π}{4}

x = - \frac{4 πn + 40 + π}{4}

x = \frac{4 πn + π - 20}{32}, x = - \frac{4 πn + 40 + π}{4}

x = \frac{4 πn + π - 20}{32}, x = - \frac{4 πn + 40 + π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(9x+15)=sin(7x-5) ?
The general solution for cos(9x+15)=sin(7x-5) is x=(4pin+pi-20}{32},x=-\frac{4pin+40+pi)/4