What is the general solution for cos(4x+4)=sin(10x+44) ?

The general solution for cos(4x+4)=sin(10x+44) is x=(4pin+pi-96)/(28),x=(pi+4pin-80)/(12)

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cos(4x+4)=sin(10x+44)

Solution

cos (4 x + 4) = sin (10 x + 44)

Solution

x = \frac{4 πn + π - 96}{28}, x = \frac{π + 4 πn - 80}{12}

Degrees

x = - 190.01410 \dots^{\circ} + 25.71428 \dots^{\circ} n, x = - 366.97186 \dots^{\circ} + 6 0^{\circ} n

Solution steps

cos (4 x + 4) = sin (10 x + 44)

Rewrite using trig identities

cos (4 x + 4) = sin (10 x + 44)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (4 x + 4) = sin (\frac{π}{2} - (4 x + 4))

cos (4 x + 4) = sin (\frac{π}{2} - (4 x + 4))

Apply trig inverse properties

cos (4 x + 4) = sin (\frac{π}{2} - (4 x + 4))

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

10 x + 44 = \frac{π}{2} - (4 x + 4) + 2 πn, 10 x + 44 = π - (\frac{π}{2} - (4 x + 4)) + 2 πn

10 x + 44 = \frac{π}{2} - (4 x + 4) + 2 πn, 10 x + 44 = π - (\frac{π}{2} - (4 x + 4)) + 2 πn

10 x + 44 = \frac{π}{2} - (4 x + 4) + 2 πn : x = \frac{4 πn + π - 96}{28}

10 x + 44 = \frac{π}{2} - (4 x + 4) + 2 πn

Expand

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

\frac{π}{2} - (4 x + 4) + 2 πn

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

- (4 x + 4)

Distribute parentheses

= - (4 x) - (4)

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 4

= \frac{π}{2} - 4 x - 4 + 2 πn

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

Move

44

to the right side

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

Subtract

44

from both sides

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

Simplify

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

Simplify

10 x + 44 - 44 : 10 x

10 x + 44 - 44

Add similar elements:

44 - 44 = 0

= 10 x

Simplify

\frac{π}{2} - 4 x - 4 + 2 πn - 44 : - 4 x + 2 πn + \frac{π}{2} - 48

\frac{π}{2} - 4 x - 4 + 2 πn - 44

Group like terms

= - 4 x + 2 πn + \frac{π}{2} - 4 - 44

Subtract the numbers:

- 4 - 44 = - 48

= - 4 x + 2 πn + \frac{π}{2} - 48

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

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

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

Move

4 x

to the left side

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

Add

4 x

to both sides

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

Simplify

14 x = 2 πn + \frac{π}{2} - 48

14 x = 2 πn + \frac{π}{2} - 48

Divide both sides by

14

14 x = 2 πn + \frac{π}{2} - 48

Divide both sides by

14

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{\frac{π}{2}}{14} - \frac{48}{14}

Simplify

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{\frac{π}{2}}{14} - \frac{48}{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{\frac{π}{2}}{14} - \frac{48}{14} : \frac{4 πn + π - 96}{28}

\frac{2 πn}{14} + \frac{\frac{π}{2}}{14} - \frac{48}{14}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 48}{14}

Join

2 πn + \frac{π}{2} - 48 : \frac{4 πn + π - 96}{2}

2 πn + \frac{π}{2} - 48

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 48 = \frac{48 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{48 \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 - 48 \cdot 2}{2}

2 πn \cdot 2 + π - 48 \cdot 2 = 4 πn + π - 96

2 πn \cdot 2 + π - 48 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 48 \cdot 2

Multiply the numbers:

48 \cdot 2 = 96

= 4 πn + π - 96

= \frac{4 πn + π - 96}{2}

= \frac{\frac{4 πn + π - 96}{2}}{14}

Apply the fraction rule:

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

= \frac{4 πn + π - 96}{2 \cdot 14}

Multiply the numbers:

2 \cdot 14 = 28

= \frac{4 πn + π - 96}{28}

x = \frac{4 πn + π - 96}{28}

x = \frac{4 πn + π - 96}{28}

x = \frac{4 πn + π - 96}{28}

10 x + 44 = π - (\frac{π}{2} - (4 x + 4)) + 2 πn : x = \frac{π + 4 πn - 80}{12}

10 x + 44 = π - (\frac{π}{2} - (4 x + 4)) + 2 πn

Expand

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

π - (\frac{π}{2} - (4 x + 4)) + 2 πn

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

- (4 x + 4)

Distribute parentheses

= - (4 x) - (4)

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 4

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

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

- (\frac{π}{2} - 4 x - 4)

Distribute parentheses

= - (\frac{π}{2}) - (- 4 x) - (- 4)

Apply minus-plus rules

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

= - \frac{π}{2} + 4 x + 4

= π - \frac{π}{2} + 4 x + 4 + 2 πn

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

Move

44

to the right side

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

Subtract

44

from both sides

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

Simplify

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

Simplify

10 x + 44 - 44 : 10 x

10 x + 44 - 44

Add similar elements:

44 - 44 = 0

= 10 x

Simplify

π - \frac{π}{2} + 4 x + 4 + 2 πn - 44 : 4 x + 2 πn + π - 40 - \frac{π}{2}

π - \frac{π}{2} + 4 x + 4 + 2 πn - 44

Group like terms

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

Add/Subtract the numbers:

4 - 44 = - 40

= 4 x + 2 πn + π - 40 - \frac{π}{2}

10 x = 4 x + 2 πn + π - 40 - \frac{π}{2}

10 x = 4 x + 2 πn + π - 40 - \frac{π}{2}

10 x = 4 x + 2 πn + π - 40 - \frac{π}{2}

Move

4 x

to the left side

10 x = 4 x + 2 πn + π - 40 - \frac{π}{2}

Subtract

4 x

from both sides

10 x - 4 x = 4 x + 2 πn + π - 40 - \frac{π}{2} - 4 x

Simplify

6 x = 2 πn + π - 40 - \frac{π}{2}

6 x = 2 πn + π - 40 - \frac{π}{2}

Divide both sides by

6

6 x = 2 πn + π - 40 - \frac{π}{2}

Divide both sides by

6

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{π}{6} - \frac{40}{6} - \frac{\frac{π}{2}}{6}

Simplify

\frac{6 x}{6} = \frac{2 πn}{6} + \frac{π}{6} - \frac{40}{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{π}{6} - \frac{40}{6} - \frac{\frac{π}{2}}{6} : \frac{π + 4 πn - 80}{12}

\frac{2 πn}{6} + \frac{π}{6} - \frac{40}{6} - \frac{\frac{π}{2}}{6}

Apply rule

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

= \frac{2 πn + π - 40 - \frac{π}{2}}{6}

Join

2 πn + π - 40 - \frac{π}{2} : \frac{π + 4 πn - 80}{2}

2 πn + π - 40 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 40 = \frac{40 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{40 \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 - 40 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 40 \cdot 2 - π = π + 4 πn - 80

2 πn \cdot 2 + π 2 - 40 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 40 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 40 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 40 \cdot 2

Multiply the numbers:

40 \cdot 2 = 80

= π + 4 πn - 80

= \frac{π + 4 πn - 80}{2}

= \frac{\frac{π + 4 πn - 80}{2}}{6}

Apply the fraction rule:

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

= \frac{π + 4 πn - 80}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π + 4 πn - 80}{12}

x = \frac{π + 4 πn - 80}{12}

x = \frac{π + 4 πn - 80}{12}

x = \frac{π + 4 πn - 80}{12}

x = \frac{4 πn + π - 96}{28}, x = \frac{π + 4 πn - 80}{12}

x = \frac{4 πn + π - 96}{28}, x = \frac{π + 4 πn - 80}{12}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(4x+4)=sin(10x+44) ?
The general solution for cos(4x+4)=sin(10x+44) is x=(4pin+pi-96)/(28),x=(pi+4pin-80)/(12)