What is the general solution for sin(5x-10)=cos(x-8) ?

The general solution for sin(5x-10)=cos(x-8) is x=(4pin+36+pi}{12},x=\frac{4pin+4+pi)/8

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sin(5x-10)=cos(x-8)

Solution

sin (5 x - 10) = cos (x - 8)

Solution

x = \frac{4 πn + 36 + π}{12}, x = \frac{4 πn + 4 + π}{8}

Degrees

x = 186.88733 \dots^{\circ} + 6 0^{\circ} n, x = 51.14788 \dots^{\circ} + 9 0^{\circ} n

Solution steps

sin (5 x - 10) = cos (x - 8)

Rewrite using trig identities

sin (5 x - 10) = cos (x - 8)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x - 10) = sin (\frac{π}{2} - (x - 8))

sin (5 x - 10) = sin (\frac{π}{2} - (x - 8))

Apply trig inverse properties

sin (5 x - 10) = sin (\frac{π}{2} - (x - 8))

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

5 x - 10 = \frac{π}{2} - (x - 8) + 2 πn, 5 x - 10 = π - (\frac{π}{2} - (x - 8)) + 2 πn

5 x - 10 = \frac{π}{2} - (x - 8) + 2 πn, 5 x - 10 = π - (\frac{π}{2} - (x - 8)) + 2 πn

5 x - 10 = \frac{π}{2} - (x - 8) + 2 πn : x = \frac{4 πn + 36 + π}{12}

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

Expand

\frac{π}{2} - (x - 8) + 2 πn : \frac{π}{2} - x + 8 + 2 πn

\frac{π}{2} - (x - 8) + 2 πn

- (x - 8) : - x + 8

- (x - 8)

Distribute parentheses

= - (x) - (- 8)

Apply minus-plus rules

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

= - x + 8

= \frac{π}{2} - x + 8 + 2 πn

5 x - 10 = \frac{π}{2} - x + 8 + 2 πn

Move

10

to the right side

5 x - 10 = \frac{π}{2} - x + 8 + 2 πn

Add

10

to both sides

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

Simplify

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

Simplify

5 x - 10 + 10 : 5 x

5 x - 10 + 10

Add similar elements:

- 10 + 10 = 0

= 5 x

Simplify

\frac{π}{2} - x + 8 + 2 πn + 10 : - x + 2 πn + 18 + \frac{π}{2}

\frac{π}{2} - x + 8 + 2 πn + 10

Group like terms

= - x + 2 πn + \frac{π}{2} + 8 + 10

Add the numbers:

8 + 10 = 18

= - x + 2 πn + 18 + \frac{π}{2}

5 x = - x + 2 πn + 18 + \frac{π}{2}

5 x = - x + 2 πn + 18 + \frac{π}{2}

5 x = - x + 2 πn + 18 + \frac{π}{2}

Move

x

to the left side

5 x = - x + 2 πn + 18 + \frac{π}{2}

Add

x

to both sides

5 x + x = - x + 2 πn + 18 + \frac{π}{2} + x

Simplify

6 x = 2 πn + 18 + \frac{π}{2}

6 x = 2 πn + 18 + \frac{π}{2}

Divide both sides by

6

6 x = 2 πn + 18 + \frac{π}{2}

Divide both sides by

6

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{18}{6} + \frac{\frac{π}{2}}{6}

Simplify

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{18}{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{18}{6} + \frac{\frac{π}{2}}{6} : \frac{4 πn + 36 + π}{12}

\frac{2 πn}{6} + \frac{18}{6} + \frac{\frac{π}{2}}{6}

Apply rule

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

= \frac{2 πn + 18 + \frac{π}{2}}{6}

Join

2 πn + 18 + \frac{π}{2} : \frac{4 πn + 36 + π}{2}

2 πn + 18 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 18 = \frac{18 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{18 \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 + 18 \cdot 2 + π}{2}

2 πn \cdot 2 + 18 \cdot 2 + π = 4 πn + 36 + π

2 πn \cdot 2 + 18 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 18 \cdot 2 + π

Multiply the numbers:

18 \cdot 2 = 36

= 4 πn + 36 + π

= \frac{4 πn + 36 + π}{2}

= \frac{\frac{4 πn + 36 + π}{2}}{6}

Apply the fraction rule:

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

= \frac{4 πn + 36 + π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4 πn + 36 + π}{12}

x = \frac{4 πn + 36 + π}{12}

x = \frac{4 πn + 36 + π}{12}

x = \frac{4 πn + 36 + π}{12}

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

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

Expand

π - (\frac{π}{2} - (x - 8)) + 2 πn : π - \frac{π}{2} + x - 8 + 2 πn

π - (\frac{π}{2} - (x - 8)) + 2 πn

- (x - 8) : - x + 8

- (x - 8)

Distribute parentheses

= - (x) - (- 8)

Apply minus-plus rules

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

= - x + 8

= π - (- x + 8 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - x + 8) : - \frac{π}{2} + x - 8

- (\frac{π}{2} - x + 8)

Distribute parentheses

= - (\frac{π}{2}) - (- x) - (8)

Apply minus-plus rules

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

= - \frac{π}{2} + x - 8

= π - \frac{π}{2} + x - 8 + 2 πn

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

Move

10

to the right side

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

Add

10

to both sides

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

Simplify

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

Simplify

5 x - 10 + 10 : 5 x

5 x - 10 + 10

Add similar elements:

- 10 + 10 = 0

= 5 x

Simplify

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

π - \frac{π}{2} + x - 8 + 2 πn + 10

Group like terms

= x + π + 2 πn - \frac{π}{2} - 8 + 10

Add/Subtract the numbers:

- 8 + 10 = 2

= x + 2 πn + 2 + π - \frac{π}{2}

5 x = x + 2 πn + 2 + π - \frac{π}{2}

5 x = x + 2 πn + 2 + π - \frac{π}{2}

5 x = x + 2 πn + 2 + π - \frac{π}{2}

Move

x

to the left side

5 x = x + 2 πn + 2 + π - \frac{π}{2}

Subtract

x

from both sides

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

Simplify

4 x = 2 πn + 2 + π - \frac{π}{2}

4 x = 2 πn + 2 + π - \frac{π}{2}

Divide both sides by

4

4 x = 2 πn + 2 + π - \frac{π}{2}

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

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

Apply rule

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

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

Join

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

2 πn + 2 + π - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + 2 \cdot 2 + π 2 - π

Add similar elements:

2 π - π = π

= 2 \cdot 2 πn + 2 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 4 + π

= \frac{4 πn + 4 + π}{2}

= \frac{\frac{4 πn + 4 + π}{2}}{4}

Apply the fraction rule:

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

= \frac{4 πn + 4 + π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4 πn + 4 + π}{8}

x = \frac{4 πn + 4 + π}{8}

x = \frac{4 πn + 4 + π}{8}

x = \frac{4 πn + 4 + π}{8}

x = \frac{4 πn + 36 + π}{12}, x = \frac{4 πn + 4 + π}{8}

x = \frac{4 πn + 36 + π}{12}, x = \frac{4 πn + 4 + π}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x-10)=cos(x-8) ?
The general solution for sin(5x-10)=cos(x-8) is x=(4pin+36+pi}{12},x=\frac{4pin+4+pi)/8