What is the general solution for sin(8x+2)=cos(6x-10) ?

The general solution for sin(8x+2)=cos(6x-10) is x=(4pin+16+pi}{28},x=\frac{pi+4pin-24)/4

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sin(8x+2)=cos(6x-10)

Solution

sin (8 x + 2) = cos (6 x - 10)

Solution

x = \frac{4 πn + 16 + π}{28}, x = \frac{π + 4 πn - 24}{4}

Degrees

x = 39.16901 \dots^{\circ} + 25.71428 \dots^{\circ} n, x = - 298.77467 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (8 x + 2) = cos (6 x - 10)

Rewrite using trig identities

sin (8 x + 2) = cos (6 x - 10)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (8 x + 2) = sin (\frac{π}{2} - (6 x - 10))

sin (8 x + 2) = sin (\frac{π}{2} - (6 x - 10))

Apply trig inverse properties

sin (8 x + 2) = sin (\frac{π}{2} - (6 x - 10))

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

8 x + 2 = \frac{π}{2} - (6 x - 10) + 2 πn, 8 x + 2 = π - (\frac{π}{2} - (6 x - 10)) + 2 πn

8 x + 2 = \frac{π}{2} - (6 x - 10) + 2 πn, 8 x + 2 = π - (\frac{π}{2} - (6 x - 10)) + 2 πn

8 x + 2 = \frac{π}{2} - (6 x - 10) + 2 πn : x = \frac{4 πn + 16 + π}{28}

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

Expand

\frac{π}{2} - (6 x - 10) + 2 πn : \frac{π}{2} - 6 x + 10 + 2 πn

\frac{π}{2} - (6 x - 10) + 2 πn

- (6 x - 10) : - 6 x + 10

- (6 x - 10)

Distribute parentheses

= - (6 x) - (- 10)

Apply minus-plus rules

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

= - 6 x + 10

= \frac{π}{2} - 6 x + 10 + 2 πn

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

Simplify

8 x + 2 - 2 : 8 x

8 x + 2 - 2

Add similar elements:

2 - 2 = 0

= 8 x

Simplify

\frac{π}{2} - 6 x + 10 + 2 πn - 2 : - 6 x + 2 πn + 8 + \frac{π}{2}

\frac{π}{2} - 6 x + 10 + 2 πn - 2

Group like terms

= - 6 x + 2 πn + \frac{π}{2} + 10 - 2

Add/Subtract the numbers:

10 - 2 = 8

= - 6 x + 2 πn + 8 + \frac{π}{2}

8 x = - 6 x + 2 πn + 8 + \frac{π}{2}

8 x = - 6 x + 2 πn + 8 + \frac{π}{2}

8 x = - 6 x + 2 πn + 8 + \frac{π}{2}

Move

6 x

to the left side

8 x = - 6 x + 2 πn + 8 + \frac{π}{2}

Add

6 x

to both sides

8 x + 6 x = - 6 x + 2 πn + 8 + \frac{π}{2} + 6 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}

8 x + 2 = π - (\frac{π}{2} - (6 x - 10)) + 2 πn : x = \frac{π + 4 πn - 24}{4}

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

Expand

π - (\frac{π}{2} - (6 x - 10)) + 2 πn : π - \frac{π}{2} + 6 x - 10 + 2 πn

π - (\frac{π}{2} - (6 x - 10)) + 2 πn

- (6 x - 10) : - 6 x + 10

- (6 x - 10)

Distribute parentheses

= - (6 x) - (- 10)

Apply minus-plus rules

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

= - 6 x + 10

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

- (\frac{π}{2} - 6 x + 10) : - \frac{π}{2} + 6 x - 10

- (\frac{π}{2} - 6 x + 10)

Distribute parentheses

= - (\frac{π}{2}) - (- 6 x) - (10)

Apply minus-plus rules

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

= - \frac{π}{2} + 6 x - 10

= π - \frac{π}{2} + 6 x - 10 + 2 πn

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

Simplify

8 x + 2 - 2 : 8 x

8 x + 2 - 2

Add similar elements:

2 - 2 = 0

= 8 x

Simplify

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

π - \frac{π}{2} + 6 x - 10 + 2 πn - 2

Group like terms

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

Subtract the numbers:

- 10 - 2 = - 12

= 6 x + 2 πn + π - 12 - \frac{π}{2}

8 x = 6 x + 2 πn + π - 12 - \frac{π}{2}

8 x = 6 x + 2 πn + π - 12 - \frac{π}{2}

8 x = 6 x + 2 πn + π - 12 - \frac{π}{2}

Move

6 x

to the left side

8 x = 6 x + 2 πn + π - 12 - \frac{π}{2}

Subtract

6 x

from both sides

8 x - 6 x = 6 x + 2 πn + π - 12 - \frac{π}{2} - 6 x

Simplify

2 x = 2 πn + π - 12 - \frac{π}{2}

2 x = 2 πn + π - 12 - \frac{π}{2}

Divide both sides by

2

2 x = 2 πn + π - 12 - \frac{π}{2}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

Apply rule

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

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

Join

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

2 πn + π - 12 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 12 = \frac{12 \cdot 2}{2}

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

2 πn \cdot 2 + π 2 - 12 \cdot 2 - π = π + 4 πn - 24

2 πn \cdot 2 + π 2 - 12 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 12 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 12 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 12 \cdot 2

Multiply the numbers:

12 \cdot 2 = 24

= π + 4 πn - 24

= \frac{π + 4 πn - 24}{2}

= \frac{\frac{π + 4 πn - 24}{2}}{2}

Apply the fraction rule:

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

= \frac{π + 4 πn - 24}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn - 24}{4}

x = \frac{π + 4 πn - 24}{4}

x = \frac{π + 4 πn - 24}{4}

x = \frac{π + 4 πn - 24}{4}

x = \frac{4 πn + 16 + π}{28}, x = \frac{π + 4 πn - 24}{4}

x = \frac{4 πn + 16 + π}{28}, x = \frac{π + 4 πn - 24}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(8x+2)=cos(6x-10) ?
The general solution for sin(8x+2)=cos(6x-10) is x=(4pin+16+pi}{28},x=\frac{pi+4pin-24)/4