What is the general solution for sin(8x+48)=cos(-4x+26) ?

The general solution for sin(8x+48)=cos(-4x+26) is x=(4pin+pi-148)/8 ,x=(pi+4pin-44)/(24)

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sin(8x+48)=cos(-4x+26)

Solution

sin (8 x + 48) = cos (- 4 x + 26)

Solution

x = \frac{4 πn + π - 148}{8}, x = \frac{π + 4 πn - 44}{24}

Degrees

x = - 1037.47192 \dots^{\circ} + 9 0^{\circ} n, x = - 97.54226 \dots^{\circ} + 3 0^{\circ} n

Solution steps

sin (8 x + 48) = cos (- 4 x + 26)

Rewrite using trig identities

sin (8 x + 48) = cos (- 4 x + 26)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (8 x + 48) = sin (\frac{π}{2} - (- 4 x + 26))

sin (8 x + 48) = sin (\frac{π}{2} - (- 4 x + 26))

Apply trig inverse properties

sin (8 x + 48) = sin (\frac{π}{2} - (- 4 x + 26))

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

8 x + 48 = \frac{π}{2} - (- 4 x + 26) + 2 πn, 8 x + 48 = π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn

8 x + 48 = \frac{π}{2} - (- 4 x + 26) + 2 πn, 8 x + 48 = π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn

8 x + 48 = \frac{π}{2} - (- 4 x + 26) + 2 πn : x = \frac{4 πn + π - 148}{8}

8 x + 48 = \frac{π}{2} - (- 4 x + 26) + 2 πn

Expand

\frac{π}{2} - (- 4 x + 26) + 2 πn : \frac{π}{2} + 4 x - 26 + 2 πn

\frac{π}{2} - (- 4 x + 26) + 2 πn

- (- 4 x + 26) : 4 x - 26

- (- 4 x + 26)

Distribute parentheses

= - (- 4 x) - (26)

Apply minus-plus rules

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

= 4 x - 26

= \frac{π}{2} + 4 x - 26 + 2 πn

8 x + 48 = \frac{π}{2} + 4 x - 26 + 2 πn

Move

48

to the right side

8 x + 48 = \frac{π}{2} + 4 x - 26 + 2 πn

Subtract

48

from both sides

8 x + 48 - 48 = \frac{π}{2} + 4 x - 26 + 2 πn - 48

Simplify

8 x + 48 - 48 = \frac{π}{2} + 4 x - 26 + 2 πn - 48

Simplify

8 x + 48 - 48 : 8 x

8 x + 48 - 48

Add similar elements:

48 - 48 = 0

= 8 x

Simplify

\frac{π}{2} + 4 x - 26 + 2 πn - 48 : 4 x + 2 πn + \frac{π}{2} - 74

\frac{π}{2} + 4 x - 26 + 2 πn - 48

Group like terms

= 4 x + 2 πn + \frac{π}{2} - 26 - 48

Subtract the numbers:

- 26 - 48 = - 74

= 4 x + 2 πn + \frac{π}{2} - 74

8 x = 4 x + 2 πn + \frac{π}{2} - 74

8 x = 4 x + 2 πn + \frac{π}{2} - 74

8 x = 4 x + 2 πn + \frac{π}{2} - 74

Move

4 x

to the left side

8 x = 4 x + 2 πn + \frac{π}{2} - 74

Subtract

4 x

from both sides

8 x - 4 x = 4 x + 2 πn + \frac{π}{2} - 74 - 4 x

Simplify

4 x = 2 πn + \frac{π}{2} - 74

4 x = 2 πn + \frac{π}{2} - 74

Divide both sides by

4

4 x = 2 πn + \frac{π}{2} - 74

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{74}{4}

Simplify

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{74}{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{\frac{π}{2}}{4} - \frac{74}{4} : \frac{4 πn + π - 148}{8}

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{74}{4}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 74}{4}

Join

2 πn + \frac{π}{2} - 74 : \frac{4 πn + π - 148}{2}

2 πn + \frac{π}{2} - 74

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 74 = \frac{74 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{74 \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 - 74 \cdot 2}{2}

2 πn \cdot 2 + π - 74 \cdot 2 = 4 πn + π - 148

2 πn \cdot 2 + π - 74 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 74 \cdot 2

Multiply the numbers:

74 \cdot 2 = 148

= 4 πn + π - 148

= \frac{4 πn + π - 148}{2}

= \frac{\frac{4 πn + π - 148}{2}}{4}

Apply the fraction rule:

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

= \frac{4 πn + π - 148}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4 πn + π - 148}{8}

x = \frac{4 πn + π - 148}{8}

x = \frac{4 πn + π - 148}{8}

x = \frac{4 πn + π - 148}{8}

8 x + 48 = π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn : x = \frac{π + 4 πn - 44}{24}

8 x + 48 = π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn

Expand

π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn : π - \frac{π}{2} - 4 x + 26 + 2 πn

π - (\frac{π}{2} - (- 4 x + 26)) + 2 πn

- (- 4 x + 26) : 4 x - 26

- (- 4 x + 26)

Distribute parentheses

= - (- 4 x) - (26)

Apply minus-plus rules

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

= 4 x - 26

= π - (4 x + \frac{π}{2} - 26) + 2 πn

- (\frac{π}{2} + 4 x - 26) : - \frac{π}{2} - 4 x + 26

- (\frac{π}{2} + 4 x - 26)

Distribute parentheses

= - (\frac{π}{2}) - (4 x) - (- 26)

Apply minus-plus rules

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

= - \frac{π}{2} - 4 x + 26

= π - \frac{π}{2} - 4 x + 26 + 2 πn

8 x + 48 = π - \frac{π}{2} - 4 x + 26 + 2 πn

Move

48

to the right side

8 x + 48 = π - \frac{π}{2} - 4 x + 26 + 2 πn

Subtract

48

from both sides

8 x + 48 - 48 = π - \frac{π}{2} - 4 x + 26 + 2 πn - 48

Simplify

8 x + 48 - 48 = π - \frac{π}{2} - 4 x + 26 + 2 πn - 48

Simplify

8 x + 48 - 48 : 8 x

8 x + 48 - 48

Add similar elements:

48 - 48 = 0

= 8 x

Simplify

π - \frac{π}{2} - 4 x + 26 + 2 πn - 48 : - 4 x + 2 πn + π - 22 - \frac{π}{2}

π - \frac{π}{2} - 4 x + 26 + 2 πn - 48

Group like terms

= - 4 x + π + 2 πn - \frac{π}{2} + 26 - 48

Add/Subtract the numbers:

26 - 48 = - 22

= - 4 x + 2 πn + π - 22 - \frac{π}{2}

8 x = - 4 x + 2 πn + π - 22 - \frac{π}{2}

8 x = - 4 x + 2 πn + π - 22 - \frac{π}{2}

8 x = - 4 x + 2 πn + π - 22 - \frac{π}{2}

Move

4 x

to the left side

8 x = - 4 x + 2 πn + π - 22 - \frac{π}{2}

Add

4 x

to both sides

8 x + 4 x = - 4 x + 2 πn + π - 22 - \frac{π}{2} + 4 x

Simplify

12 x = 2 πn + π - 22 - \frac{π}{2}

12 x = 2 πn + π - 22 - \frac{π}{2}

Divide both sides by

12

12 x = 2 πn + π - 22 - \frac{π}{2}

Divide both sides by

12

\frac{12 x}{12} = \frac{2 πn}{12} + \frac{π}{12} - \frac{22}{12} - \frac{\frac{π}{2}}{12}

Simplify

\frac{12 x}{12} = \frac{2 πn}{12} + \frac{π}{12} - \frac{22}{12} - \frac{\frac{π}{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{π}{12} - \frac{22}{12} - \frac{\frac{π}{2}}{12} : \frac{π + 4 πn - 44}{24}

\frac{2 πn}{12} + \frac{π}{12} - \frac{22}{12} - \frac{\frac{π}{2}}{12}

Apply rule

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

= \frac{2 πn + π - 22 - \frac{π}{2}}{12}

Join

2 πn + π - 22 - \frac{π}{2} : \frac{π + 4 πn - 44}{2}

2 πn + π - 22 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 22 = \frac{22 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{22 \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 - 22 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 22 \cdot 2 - π = π + 4 πn - 44

2 πn \cdot 2 + π 2 - 22 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 22 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 22 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 22 \cdot 2

Multiply the numbers:

22 \cdot 2 = 44

= π + 4 πn - 44

= \frac{π + 4 πn - 44}{2}

= \frac{\frac{π + 4 πn - 44}{2}}{12}

Apply the fraction rule:

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

= \frac{π + 4 πn - 44}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{π + 4 πn - 44}{24}

x = \frac{π + 4 πn - 44}{24}

x = \frac{π + 4 πn - 44}{24}

x = \frac{π + 4 πn - 44}{24}

x = \frac{4 πn + π - 148}{8}, x = \frac{π + 4 πn - 44}{24}

x = \frac{4 πn + π - 148}{8}, x = \frac{π + 4 πn - 44}{24}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(8x+48)=cos(-4x+26) ?
The general solution for sin(8x+48)=cos(-4x+26) is x=(4pin+pi-148)/8 ,x=(pi+4pin-44)/(24)