What is the general solution for sin(5x+50)=cos(4x-23) ?

The general solution for sin(5x+50)=cos(4x-23) is x=(4pin+pi-54)/(18),x=2pin+pi-73-pi/2

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Popular Trigonometry >

sin(5x+50)=cos(4x-23)

Solution

sin (5 x + 50) = cos (4 x - 23)

Solution

x = \frac{4 πn + π - 54}{18}, x = 2 πn + π - 73 - \frac{π}{2}

Degrees

x = - 161.88733 \dots^{\circ} + 4 0^{\circ} n, x = - 4092.59190 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (5 x + 50) = cos (4 x - 23)

Rewrite using trig identities

sin (5 x + 50) = cos (4 x - 23)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x + 50) = sin (\frac{π}{2} - (4 x - 23))

sin (5 x + 50) = sin (\frac{π}{2} - (4 x - 23))

Apply trig inverse properties

sin (5 x + 50) = sin (\frac{π}{2} - (4 x - 23))

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

5 x + 50 = \frac{π}{2} - (4 x - 23) + 2 πn, 5 x + 50 = π - (\frac{π}{2} - (4 x - 23)) + 2 πn

5 x + 50 = \frac{π}{2} - (4 x - 23) + 2 πn, 5 x + 50 = π - (\frac{π}{2} - (4 x - 23)) + 2 πn

5 x + 50 = \frac{π}{2} - (4 x - 23) + 2 πn : x = \frac{4 πn + π - 54}{18}

5 x + 50 = \frac{π}{2} - (4 x - 23) + 2 πn

Expand

\frac{π}{2} - (4 x - 23) + 2 πn : \frac{π}{2} - 4 x + 23 + 2 πn

\frac{π}{2} - (4 x - 23) + 2 πn

- (4 x - 23) : - 4 x + 23

- (4 x - 23)

Distribute parentheses

= - (4 x) - (- 23)

Apply minus-plus rules

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

= - 4 x + 23

= \frac{π}{2} - 4 x + 23 + 2 πn

5 x + 50 = \frac{π}{2} - 4 x + 23 + 2 πn

Move

50

to the right side

5 x + 50 = \frac{π}{2} - 4 x + 23 + 2 πn

Subtract

50

from both sides

5 x + 50 - 50 = \frac{π}{2} - 4 x + 23 + 2 πn - 50

Simplify

5 x + 50 - 50 = \frac{π}{2} - 4 x + 23 + 2 πn - 50

Simplify

5 x + 50 - 50 : 5 x

5 x + 50 - 50

Add similar elements:

50 - 50 = 0

= 5 x

Simplify

\frac{π}{2} - 4 x + 23 + 2 πn - 50 : - 4 x + 2 πn + \frac{π}{2} - 27

\frac{π}{2} - 4 x + 23 + 2 πn - 50

Group like terms

= - 4 x + 2 πn + \frac{π}{2} + 23 - 50

Add/Subtract the numbers:

23 - 50 = - 27

= - 4 x + 2 πn + \frac{π}{2} - 27

5 x = - 4 x + 2 πn + \frac{π}{2} - 27

5 x = - 4 x + 2 πn + \frac{π}{2} - 27

5 x = - 4 x + 2 πn + \frac{π}{2} - 27

Move

4 x

to the left side

5 x = - 4 x + 2 πn + \frac{π}{2} - 27

Add

4 x

to both sides

5 x + 4 x = - 4 x + 2 πn + \frac{π}{2} - 27 + 4 x

Simplify

9 x = 2 πn + \frac{π}{2} - 27

9 x = 2 πn + \frac{π}{2} - 27

Divide both sides by

9

9 x = 2 πn + \frac{π}{2} - 27

Divide both sides by

9

\frac{9 x}{9} = \frac{2 πn}{9} + \frac{\frac{π}{2}}{9} - \frac{27}{9}

Simplify

\frac{9 x}{9} = \frac{2 πn}{9} + \frac{\frac{π}{2}}{9} - \frac{27}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{2 πn}{9} + \frac{\frac{π}{2}}{9} - \frac{27}{9} : \frac{4 πn + π - 54}{18}

\frac{2 πn}{9} + \frac{\frac{π}{2}}{9} - \frac{27}{9}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 27}{9}

Join

2 πn + \frac{π}{2} - 27 : \frac{4 πn + π - 54}{2}

2 πn + \frac{π}{2} - 27

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 27 = \frac{27 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{27 \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 - 27 \cdot 2}{2}

2 πn \cdot 2 + π - 27 \cdot 2 = 4 πn + π - 54

2 πn \cdot 2 + π - 27 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 27 \cdot 2

Multiply the numbers:

27 \cdot 2 = 54

= 4 πn + π - 54

= \frac{4 πn + π - 54}{2}

= \frac{\frac{4 πn + π - 54}{2}}{9}

Apply the fraction rule:

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

= \frac{4 πn + π - 54}{2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{4 πn + π - 54}{18}

x = \frac{4 πn + π - 54}{18}

x = \frac{4 πn + π - 54}{18}

x = \frac{4 πn + π - 54}{18}

5 x + 50 = π - (\frac{π}{2} - (4 x - 23)) + 2 πn : x = 2 πn + π - 73 - \frac{π}{2}

5 x + 50 = π - (\frac{π}{2} - (4 x - 23)) + 2 πn

Expand

π - (\frac{π}{2} - (4 x - 23)) + 2 πn : π - \frac{π}{2} + 4 x - 23 + 2 πn

π - (\frac{π}{2} - (4 x - 23)) + 2 πn

- (4 x - 23) : - 4 x + 23

- (4 x - 23)

Distribute parentheses

= - (4 x) - (- 23)

Apply minus-plus rules

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

= - 4 x + 23

= π - (- 4 x + 23 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 4 x + 23) : - \frac{π}{2} + 4 x - 23

- (\frac{π}{2} - 4 x + 23)

Distribute parentheses

= - (\frac{π}{2}) - (- 4 x) - (23)

Apply minus-plus rules

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

= - \frac{π}{2} + 4 x - 23

= π - \frac{π}{2} + 4 x - 23 + 2 πn

5 x + 50 = π - \frac{π}{2} + 4 x - 23 + 2 πn

Move

50

to the right side

5 x + 50 = π - \frac{π}{2} + 4 x - 23 + 2 πn

Subtract

50

from both sides

5 x + 50 - 50 = π - \frac{π}{2} + 4 x - 23 + 2 πn - 50

Simplify

5 x + 50 - 50 = π - \frac{π}{2} + 4 x - 23 + 2 πn - 50

Simplify

5 x + 50 - 50 : 5 x

5 x + 50 - 50

Add similar elements:

50 - 50 = 0

= 5 x

Simplify

π - \frac{π}{2} + 4 x - 23 + 2 πn - 50 : 4 x + 2 πn + π - 73 - \frac{π}{2}

π - \frac{π}{2} + 4 x - 23 + 2 πn - 50

Group like terms

= 4 x + π + 2 πn - \frac{π}{2} - 23 - 50

Subtract the numbers:

- 23 - 50 = - 73

= 4 x + 2 πn + π - 73 - \frac{π}{2}

5 x = 4 x + 2 πn + π - 73 - \frac{π}{2}

5 x = 4 x + 2 πn + π - 73 - \frac{π}{2}

5 x = 4 x + 2 πn + π - 73 - \frac{π}{2}

Move

4 x

to the left side

5 x = 4 x + 2 πn + π - 73 - \frac{π}{2}

Subtract

4 x

from both sides

5 x - 4 x = 4 x + 2 πn + π - 73 - \frac{π}{2} - 4 x

Simplify

x = 2 πn + π - 73 - \frac{π}{2}

x = 2 πn + π - 73 - \frac{π}{2}

x = \frac{4 πn + π - 54}{18}, x = 2 πn + π - 73 - \frac{π}{2}

x = \frac{4 πn + π - 54}{18}, x = 2 πn + π - 73 - \frac{π}{2}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x+50)=cos(4x-23) ?
The general solution for sin(5x+50)=cos(4x-23) is x=(4pin+pi-54)/(18),x=2pin+pi-73-pi/2