What is the general solution for sin(9x+2)=cos(6x-7) ?

The general solution for sin(9x+2)=cos(6x-7) is x=(4pin+10+pi}{30},x=\frac{pi+4pin-18)/6

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sin(9x+2)=cos(6x-7)

Solution

sin (9 x + 2) = cos (6 x - 7)

Solution

x = \frac{4 πn + 10 + π}{30}, x = \frac{π + 4 πn - 18}{6}

Degrees

x = 25.09859 \dots^{\circ} + 2 4^{\circ} n, x = - 141.88733 \dots^{\circ} + 12 0^{\circ} n

Solution steps

sin (9 x + 2) = cos (6 x - 7)

Rewrite using trig identities

sin (9 x + 2) = cos (6 x - 7)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (9 x + 2) = sin (\frac{π}{2} - (6 x - 7))

sin (9 x + 2) = sin (\frac{π}{2} - (6 x - 7))

Apply trig inverse properties

sin (9 x + 2) = sin (\frac{π}{2} - (6 x - 7))

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

9 x + 2 = \frac{π}{2} - (6 x - 7) + 2 πn, 9 x + 2 = π - (\frac{π}{2} - (6 x - 7)) + 2 πn

9 x + 2 = \frac{π}{2} - (6 x - 7) + 2 πn, 9 x + 2 = π - (\frac{π}{2} - (6 x - 7)) + 2 πn

9 x + 2 = \frac{π}{2} - (6 x - 7) + 2 πn : x = \frac{4 πn + 10 + π}{30}

9 x + 2 = \frac{π}{2} - (6 x - 7) + 2 πn

Expand

\frac{π}{2} - (6 x - 7) + 2 πn : \frac{π}{2} - 6 x + 7 + 2 πn

\frac{π}{2} - (6 x - 7) + 2 πn

- (6 x - 7) : - 6 x + 7

- (6 x - 7)

Distribute parentheses

= - (6 x) - (- 7)

Apply minus-plus rules

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

= - 6 x + 7

= \frac{π}{2} - 6 x + 7 + 2 πn

9 x + 2 = \frac{π}{2} - 6 x + 7 + 2 πn

Move

2

to the right side

9 x + 2 = \frac{π}{2} - 6 x + 7 + 2 πn

Subtract

2

from both sides

9 x + 2 - 2 = \frac{π}{2} - 6 x + 7 + 2 πn - 2

Simplify

9 x + 2 - 2 = \frac{π}{2} - 6 x + 7 + 2 πn - 2

Simplify

9 x + 2 - 2 : 9 x

9 x + 2 - 2

Add similar elements:

2 - 2 = 0

= 9 x

Simplify

\frac{π}{2} - 6 x + 7 + 2 πn - 2 : - 6 x + 2 πn + 5 + \frac{π}{2}

\frac{π}{2} - 6 x + 7 + 2 πn - 2

Group like terms

= - 6 x + 2 πn + \frac{π}{2} + 7 - 2

Add/Subtract the numbers:

7 - 2 = 5

= - 6 x + 2 πn + 5 + \frac{π}{2}

9 x = - 6 x + 2 πn + 5 + \frac{π}{2}

9 x = - 6 x + 2 πn + 5 + \frac{π}{2}

9 x = - 6 x + 2 πn + 5 + \frac{π}{2}

Move

6 x

to the left side

9 x = - 6 x + 2 πn + 5 + \frac{π}{2}

Add

6 x

to both sides

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

Simplify

15 x = 2 πn + 5 + \frac{π}{2}

15 x = 2 πn + 5 + \frac{π}{2}

Divide both sides by

15

15 x = 2 πn + 5 + \frac{π}{2}

Divide both sides by

15

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

Simplify

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

Simplify

\frac{15 x}{15} : x

\frac{15 x}{15}

Divide the numbers:

\frac{15}{15} = 1

= x

Simplify

\frac{2 πn}{15} + \frac{5}{15} + \frac{\frac{π}{2}}{15} : \frac{4 πn + 10 + π}{30}

\frac{2 πn}{15} + \frac{5}{15} + \frac{\frac{π}{2}}{15}

Apply rule

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

= \frac{2 πn + 5 + \frac{π}{2}}{15}

Join

2 πn + 5 + \frac{π}{2} : \frac{4 πn + 10 + π}{2}

2 πn + 5 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 5 = \frac{5 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{5 \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 + 5 \cdot 2 + π}{2}

2 πn \cdot 2 + 5 \cdot 2 + π = 4 πn + 10 + π

2 πn \cdot 2 + 5 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 5 \cdot 2 + π

Multiply the numbers:

5 \cdot 2 = 10

= 4 πn + 10 + π

= \frac{4 πn + 10 + π}{2}

= \frac{\frac{4 πn + 10 + π}{2}}{15}

Apply the fraction rule:

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

= \frac{4 πn + 10 + π}{2 \cdot 15}

Multiply the numbers:

2 \cdot 15 = 30

= \frac{4 πn + 10 + π}{30}

x = \frac{4 πn + 10 + π}{30}

x = \frac{4 πn + 10 + π}{30}

x = \frac{4 πn + 10 + π}{30}

9 x + 2 = π - (\frac{π}{2} - (6 x - 7)) + 2 πn : x = \frac{π + 4 πn - 18}{6}

9 x + 2 = π - (\frac{π}{2} - (6 x - 7)) + 2 πn

Expand

π - (\frac{π}{2} - (6 x - 7)) + 2 πn : π - \frac{π}{2} + 6 x - 7 + 2 πn

π - (\frac{π}{2} - (6 x - 7)) + 2 πn

- (6 x - 7) : - 6 x + 7

- (6 x - 7)

Distribute parentheses

= - (6 x) - (- 7)

Apply minus-plus rules

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

= - 6 x + 7

= π - (- 6 x + 7 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 6 x + 7) : - \frac{π}{2} + 6 x - 7

- (\frac{π}{2} - 6 x + 7)

Distribute parentheses

= - (\frac{π}{2}) - (- 6 x) - (7)

Apply minus-plus rules

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

= - \frac{π}{2} + 6 x - 7

= π - \frac{π}{2} + 6 x - 7 + 2 πn

9 x + 2 = π - \frac{π}{2} + 6 x - 7 + 2 πn

Move

2

to the right side

9 x + 2 = π - \frac{π}{2} + 6 x - 7 + 2 πn

Subtract

2

from both sides

9 x + 2 - 2 = π - \frac{π}{2} + 6 x - 7 + 2 πn - 2

Simplify

9 x + 2 - 2 = π - \frac{π}{2} + 6 x - 7 + 2 πn - 2

Simplify

9 x + 2 - 2 : 9 x

9 x + 2 - 2

Add similar elements:

2 - 2 = 0

= 9 x

Simplify

π - \frac{π}{2} + 6 x - 7 + 2 πn - 2 : 6 x + 2 πn + π - 9 - \frac{π}{2}

π - \frac{π}{2} + 6 x - 7 + 2 πn - 2

Group like terms

= 6 x + π + 2 πn - \frac{π}{2} - 7 - 2

Subtract the numbers:

- 7 - 2 = - 9

= 6 x + 2 πn + π - 9 - \frac{π}{2}

9 x = 6 x + 2 πn + π - 9 - \frac{π}{2}

9 x = 6 x + 2 πn + π - 9 - \frac{π}{2}

9 x = 6 x + 2 πn + π - 9 - \frac{π}{2}

Move

6 x

to the left side

9 x = 6 x + 2 πn + π - 9 - \frac{π}{2}

Subtract

6 x

from both sides

9 x - 6 x = 6 x + 2 πn + π - 9 - \frac{π}{2} - 6 x

Simplify

3 x = 2 πn + π - 9 - \frac{π}{2}

3 x = 2 πn + π - 9 - \frac{π}{2}

Divide both sides by

3

3 x = 2 πn + π - 9 - \frac{π}{2}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{π}{3} - \frac{9}{3} - \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{π}{3} - \frac{9}{3} - \frac{\frac{π}{2}}{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{π}{3} - \frac{9}{3} - \frac{\frac{π}{2}}{3} : \frac{π + 4 πn - 18}{6}

\frac{2 πn}{3} + \frac{π}{3} - \frac{9}{3} - \frac{\frac{π}{2}}{3}

Apply rule

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

= \frac{2 πn + π - 9 - \frac{π}{2}}{3}

Join

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

2 πn + π - 9 - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + π 2 - 9 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 9 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 9 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 9 \cdot 2

Multiply the numbers:

9 \cdot 2 = 18

= π + 4 πn - 18

= \frac{π + 4 πn - 18}{2}

= \frac{\frac{π + 4 πn - 18}{2}}{3}

Apply the fraction rule:

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

= \frac{π + 4 πn - 18}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 4 πn - 18}{6}

x = \frac{π + 4 πn - 18}{6}

x = \frac{π + 4 πn - 18}{6}

x = \frac{π + 4 πn - 18}{6}

x = \frac{4 πn + 10 + π}{30}, x = \frac{π + 4 πn - 18}{6}

x = \frac{4 πn + 10 + π}{30}, x = \frac{π + 4 πn - 18}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(9x+2)=cos(6x-7) ?
The general solution for sin(9x+2)=cos(6x-7) is x=(4pin+10+pi}{30},x=\frac{pi+4pin-18)/6