What is the general solution for sin(9x+6)=cos(3x-4) ?

The general solution for sin(9x+6)=cos(3x-4) is x=(4pin+pi-4)/(24),x=(pi+4pin-20)/(12)

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sin(9x+6)=cos(3x-4)

Solution

sin (9 x + 6) = cos (3 x - 4)

Solution

x = \frac{4 πn + π - 4}{24}, x = \frac{π + 4 πn - 20}{12}

Degrees

x = - 2.04929 \dots^{\circ} + 3 0^{\circ} n, x = - 80.49296 \dots^{\circ} + 6 0^{\circ} n

Solution steps

sin (9 x + 6) = cos (3 x - 4)

Rewrite using trig identities

sin (9 x + 6) = cos (3 x - 4)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (9 x + 6) = sin (\frac{π}{2} - (3 x - 4))

sin (9 x + 6) = sin (\frac{π}{2} - (3 x - 4))

Apply trig inverse properties

sin (9 x + 6) = sin (\frac{π}{2} - (3 x - 4))

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

9 x + 6 = \frac{π}{2} - (3 x - 4) + 2 πn, 9 x + 6 = π - (\frac{π}{2} - (3 x - 4)) + 2 πn

9 x + 6 = \frac{π}{2} - (3 x - 4) + 2 πn, 9 x + 6 = π - (\frac{π}{2} - (3 x - 4)) + 2 πn

9 x + 6 = \frac{π}{2} - (3 x - 4) + 2 πn : x = \frac{4 πn + π - 4}{24}

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

Expand

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

\frac{π}{2} - (3 x - 4) + 2 πn

- (3 x - 4) : - 3 x + 4

- (3 x - 4)

Distribute parentheses

= - (3 x) - (- 4)

Apply minus-plus rules

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

= - 3 x + 4

= \frac{π}{2} - 3 x + 4 + 2 πn

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

Move

6

to the right side

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

Subtract

6

from both sides

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

Simplify

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

Simplify

9 x + 6 - 6 : 9 x

9 x + 6 - 6

Add similar elements:

6 - 6 = 0

= 9 x

Simplify

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

\frac{π}{2} - 3 x + 4 + 2 πn - 6

Group like terms

= - 3 x + 2 πn + \frac{π}{2} + 4 - 6

Add/Subtract the numbers:

4 - 6 = - 2

= - 3 x + 2 πn + \frac{π}{2} - 2

9 x = - 3 x + 2 πn + \frac{π}{2} - 2

9 x = - 3 x + 2 πn + \frac{π}{2} - 2

9 x = - 3 x + 2 πn + \frac{π}{2} - 2

Move

3 x

to the left side

9 x = - 3 x + 2 πn + \frac{π}{2} - 2

Add

3 x

to both sides

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

Simplify

12 x = 2 πn + \frac{π}{2} - 2

12 x = 2 πn + \frac{π}{2} - 2

Divide both sides by

12

12 x = 2 πn + \frac{π}{2} - 2

Divide both sides by

12

\frac{12 x}{12} = \frac{2 πn}{12} + \frac{\frac{π}{2}}{12} - \frac{2}{12}

Simplify

\frac{12 x}{12} = \frac{2 πn}{12} + \frac{\frac{π}{2}}{12} - \frac{2}{12}

Simplify

\frac{12 x}{12} : x

\frac{12 x}{12}

Divide the numbers:

\frac{12}{12} = 1

= x

Simplify

\frac{2 πn}{12} + \frac{\frac{π}{2}}{12} - \frac{2}{12} : \frac{4 πn + π - 4}{24}

\frac{2 πn}{12} + \frac{\frac{π}{2}}{12} - \frac{2}{12}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 2}{12}

Join

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

2 πn + \frac{π}{2} - 2

Convert element to fraction:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 πn + π - 4}{2}

= \frac{\frac{4 πn + π - 4}{2}}{12}

Apply the fraction rule:

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

= \frac{4 πn + π - 4}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{4 πn + π - 4}{24}

x = \frac{4 πn + π - 4}{24}

x = \frac{4 πn + π - 4}{24}

x = \frac{4 πn + π - 4}{24}

9 x + 6 = π - (\frac{π}{2} - (3 x - 4)) + 2 πn : x = \frac{π + 4 πn - 20}{12}

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

Expand

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

π - (\frac{π}{2} - (3 x - 4)) + 2 πn

- (3 x - 4) : - 3 x + 4

- (3 x - 4)

Distribute parentheses

= - (3 x) - (- 4)

Apply minus-plus rules

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

= - 3 x + 4

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

- (\frac{π}{2} - 3 x + 4) : - \frac{π}{2} + 3 x - 4

- (\frac{π}{2} - 3 x + 4)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x) - (4)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x - 4

= π - \frac{π}{2} + 3 x - 4 + 2 πn

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

Move

6

to the right side

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

Subtract

6

from both sides

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

Simplify

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

Simplify

9 x + 6 - 6 : 9 x

9 x + 6 - 6

Add similar elements:

6 - 6 = 0

= 9 x

Simplify

π - \frac{π}{2} + 3 x - 4 + 2 πn - 6 : 3 x + 2 πn + π - 10 - \frac{π}{2}

π - \frac{π}{2} + 3 x - 4 + 2 πn - 6

Group like terms

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

Subtract the numbers:

- 4 - 6 = - 10

= 3 x + 2 πn + π - 10 - \frac{π}{2}

9 x = 3 x + 2 πn + π - 10 - \frac{π}{2}

9 x = 3 x + 2 πn + π - 10 - \frac{π}{2}

9 x = 3 x + 2 πn + π - 10 - \frac{π}{2}

Move

3 x

to the left side

9 x = 3 x + 2 πn + π - 10 - \frac{π}{2}

Subtract

3 x

from both sides

9 x - 3 x = 3 x + 2 πn + π - 10 - \frac{π}{2} - 3 x

Simplify

6 x = 2 πn + π - 10 - \frac{π}{2}

6 x = 2 πn + π - 10 - \frac{π}{2}

Divide both sides by

6

6 x = 2 πn + π - 10 - \frac{π}{2}

Divide both sides by

6

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{π}{6} - \frac{10}{6} - \frac{\frac{π}{2}}{6}

Simplify

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{π}{6} - \frac{10}{6} - \frac{\frac{π}{2}}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{2 πn}{6} + \frac{π}{6} - \frac{10}{6} - \frac{\frac{π}{2}}{6} : \frac{π + 4 πn - 20}{12}

\frac{2 πn}{6} + \frac{π}{6} - \frac{10}{6} - \frac{\frac{π}{2}}{6}

Apply rule

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

= \frac{2 πn + π - 10 - \frac{π}{2}}{6}

Join

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

2 πn + π - 10 - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + π 2 - 10 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 10 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 10 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 10 \cdot 2

Multiply the numbers:

10 \cdot 2 = 20

= π + 4 πn - 20

= \frac{π + 4 πn - 20}{2}

= \frac{\frac{π + 4 πn - 20}{2}}{6}

Apply the fraction rule:

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

= \frac{π + 4 πn - 20}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π + 4 πn - 20}{12}

x = \frac{π + 4 πn - 20}{12}

x = \frac{π + 4 πn - 20}{12}

x = \frac{π + 4 πn - 20}{12}

x = \frac{4 πn + π - 4}{24}, x = \frac{π + 4 πn - 20}{12}

x = \frac{4 πn + π - 4}{24}, x = \frac{π + 4 πn - 20}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(9x+6)=cos(3x-4) ?
The general solution for sin(9x+6)=cos(3x-4) is x=(4pin+pi-4)/(24),x=(pi+4pin-20)/(12)