What is the general solution for sin(5x-28)=cos(3x-50) ?

The general solution for sin(5x-28)=cos(3x-50) is x=(4pin+156+pi}{16},x=\frac{pi+4pin-44)/4

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sin(5x-28)=cos(3x-50)

Solution

sin (5 x - 28) = cos (3 x - 50)

Solution

x = \frac{4 πn + 156 + π}{16}, x = \frac{π + 4 πn - 44}{4}

Degrees

x = 569.88385 \dots^{\circ} + 4 5^{\circ} n, x = - 585.25357 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (5 x - 28) = cos (3 x - 50)

Rewrite using trig identities

sin (5 x - 28) = cos (3 x - 50)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x - 28) = sin (\frac{π}{2} - (3 x - 50))

sin (5 x - 28) = sin (\frac{π}{2} - (3 x - 50))

Apply trig inverse properties

sin (5 x - 28) = sin (\frac{π}{2} - (3 x - 50))

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

5 x - 28 = \frac{π}{2} - (3 x - 50) + 2 πn, 5 x - 28 = π - (\frac{π}{2} - (3 x - 50)) + 2 πn

5 x - 28 = \frac{π}{2} - (3 x - 50) + 2 πn, 5 x - 28 = π - (\frac{π}{2} - (3 x - 50)) + 2 πn

5 x - 28 = \frac{π}{2} - (3 x - 50) + 2 πn : x = \frac{4 πn + 156 + π}{16}

5 x - 28 = \frac{π}{2} - (3 x - 50) + 2 πn

Expand

\frac{π}{2} - (3 x - 50) + 2 πn : \frac{π}{2} - 3 x + 50 + 2 πn

\frac{π}{2} - (3 x - 50) + 2 πn

- (3 x - 50) : - 3 x + 50

- (3 x - 50)

Distribute parentheses

= - (3 x) - (- 50)

Apply minus-plus rules

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

= - 3 x + 50

= \frac{π}{2} - 3 x + 50 + 2 πn

5 x - 28 = \frac{π}{2} - 3 x + 50 + 2 πn

Move

28

to the right side

5 x - 28 = \frac{π}{2} - 3 x + 50 + 2 πn

Add

28

to both sides

5 x - 28 + 28 = \frac{π}{2} - 3 x + 50 + 2 πn + 28

Simplify

5 x - 28 + 28 = \frac{π}{2} - 3 x + 50 + 2 πn + 28

Simplify

5 x - 28 + 28 : 5 x

5 x - 28 + 28

Add similar elements:

- 28 + 28 = 0

= 5 x

Simplify

\frac{π}{2} - 3 x + 50 + 2 πn + 28 : - 3 x + 2 πn + 78 + \frac{π}{2}

\frac{π}{2} - 3 x + 50 + 2 πn + 28

Group like terms

= - 3 x + 2 πn + \frac{π}{2} + 50 + 28

Add the numbers:

50 + 28 = 78

= - 3 x + 2 πn + 78 + \frac{π}{2}

5 x = - 3 x + 2 πn + 78 + \frac{π}{2}

5 x = - 3 x + 2 πn + 78 + \frac{π}{2}

5 x = - 3 x + 2 πn + 78 + \frac{π}{2}

Move

3 x

to the left side

5 x = - 3 x + 2 πn + 78 + \frac{π}{2}

Add

3 x

to both sides

5 x + 3 x = - 3 x + 2 πn + 78 + \frac{π}{2} + 3 x

Simplify

8 x = 2 πn + 78 + \frac{π}{2}

8 x = 2 πn + 78 + \frac{π}{2}

Divide both sides by

8

8 x = 2 πn + 78 + \frac{π}{2}

Divide both sides by

8

\frac{8 x}{8} = \frac{2 πn}{8} + \frac{78}{8} + \frac{\frac{π}{2}}{8}

Simplify

\frac{8 x}{8} = \frac{2 πn}{8} + \frac{78}{8} + \frac{\frac{π}{2}}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{2 πn}{8} + \frac{78}{8} + \frac{\frac{π}{2}}{8} : \frac{4 πn + 156 + π}{16}

\frac{2 πn}{8} + \frac{78}{8} + \frac{\frac{π}{2}}{8}

Apply rule

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

= \frac{2 πn + 78 + \frac{π}{2}}{8}

Join

2 πn + 78 + \frac{π}{2} : \frac{4 πn + 156 + π}{2}

2 πn + 78 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 78 = \frac{78 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{78 \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 + 78 \cdot 2 + π}{2}

2 πn \cdot 2 + 78 \cdot 2 + π = 4 πn + 156 + π

2 πn \cdot 2 + 78 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 78 \cdot 2 + π

Multiply the numbers:

78 \cdot 2 = 156

= 4 πn + 156 + π

= \frac{4 πn + 156 + π}{2}

= \frac{\frac{4 πn + 156 + π}{2}}{8}

Apply the fraction rule:

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

= \frac{4 πn + 156 + π}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{4 πn + 156 + π}{16}

x = \frac{4 πn + 156 + π}{16}

x = \frac{4 πn + 156 + π}{16}

x = \frac{4 πn + 156 + π}{16}

5 x - 28 = π - (\frac{π}{2} - (3 x - 50)) + 2 πn : x = \frac{π + 4 πn - 44}{4}

5 x - 28 = π - (\frac{π}{2} - (3 x - 50)) + 2 πn

Expand

π - (\frac{π}{2} - (3 x - 50)) + 2 πn : π - \frac{π}{2} + 3 x - 50 + 2 πn

π - (\frac{π}{2} - (3 x - 50)) + 2 πn

- (3 x - 50) : - 3 x + 50

- (3 x - 50)

Distribute parentheses

= - (3 x) - (- 50)

Apply minus-plus rules

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

= - 3 x + 50

= π - (- 3 x + 50 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 3 x + 50) : - \frac{π}{2} + 3 x - 50

- (\frac{π}{2} - 3 x + 50)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x) - (50)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x - 50

= π - \frac{π}{2} + 3 x - 50 + 2 πn

5 x - 28 = π - \frac{π}{2} + 3 x - 50 + 2 πn

Move

28

to the right side

5 x - 28 = π - \frac{π}{2} + 3 x - 50 + 2 πn

Add

28

to both sides

5 x - 28 + 28 = π - \frac{π}{2} + 3 x - 50 + 2 πn + 28

Simplify

5 x - 28 + 28 = π - \frac{π}{2} + 3 x - 50 + 2 πn + 28

Simplify

5 x - 28 + 28 : 5 x

5 x - 28 + 28

Add similar elements:

- 28 + 28 = 0

= 5 x

Simplify

π - \frac{π}{2} + 3 x - 50 + 2 πn + 28 : 3 x + 2 πn + π - 22 - \frac{π}{2}

π - \frac{π}{2} + 3 x - 50 + 2 πn + 28

Group like terms

= 3 x + π + 2 πn - \frac{π}{2} - 50 + 28

Add/Subtract the numbers:

- 50 + 28 = - 22

= 3 x + 2 πn + π - 22 - \frac{π}{2}

5 x = 3 x + 2 πn + π - 22 - \frac{π}{2}

5 x = 3 x + 2 πn + π - 22 - \frac{π}{2}

5 x = 3 x + 2 πn + π - 22 - \frac{π}{2}

Move

3 x

to the left side

5 x = 3 x + 2 πn + π - 22 - \frac{π}{2}

Subtract

3 x

from both sides

5 x - 3 x = 3 x + 2 πn + π - 22 - \frac{π}{2} - 3 x

Simplify

2 x = 2 πn + π - 22 - \frac{π}{2}

2 x = 2 πn + π - 22 - \frac{π}{2}

Divide both sides by

2

2 x = 2 πn + π - 22 - \frac{π}{2}

Divide both sides by

2

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

Simplify

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

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

Apply rule

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

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

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}}{2}

Apply the fraction rule:

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

= \frac{π + 4 πn - 44}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn - 44}{4}

x = \frac{π + 4 πn - 44}{4}

x = \frac{π + 4 πn - 44}{4}

x = \frac{π + 4 πn - 44}{4}

x = \frac{4 πn + 156 + π}{16}, x = \frac{π + 4 πn - 44}{4}

x = \frac{4 πn + 156 + π}{16}, x = \frac{π + 4 πn - 44}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x-28)=cos(3x-50) ?
The general solution for sin(5x-28)=cos(3x-50) is x=(4pin+156+pi}{16},x=\frac{pi+4pin-44)/4