What is the general solution for sin(3x+5)=cos(4x+1) ?

The general solution for sin(3x+5)=cos(4x+1) is x=(4pin+pi-12}{14},x=-\frac{pi+4pin-8)/2

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sin(3x+5)=cos(4x+1)

Solution

sin (3 x + 5) = cos (4 x + 1)

Solution

x = \frac{4 πn + π - 12}{14}, x = - \frac{π + 4 πn - 8}{2}

Degrees

x = - 36.25352 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = 139.18311 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (3 x + 5) = cos (4 x + 1)

Rewrite using trig identities

sin (3 x + 5) = cos (4 x + 1)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (3 x + 5) = sin (\frac{π}{2} - (4 x + 1))

sin (3 x + 5) = sin (\frac{π}{2} - (4 x + 1))

Apply trig inverse properties

sin (3 x + 5) = sin (\frac{π}{2} - (4 x + 1))

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

3 x + 5 = \frac{π}{2} - (4 x + 1) + 2 πn, 3 x + 5 = π - (\frac{π}{2} - (4 x + 1)) + 2 πn

3 x + 5 = \frac{π}{2} - (4 x + 1) + 2 πn, 3 x + 5 = π - (\frac{π}{2} - (4 x + 1)) + 2 πn

3 x + 5 = \frac{π}{2} - (4 x + 1) + 2 πn : x = \frac{4 πn + π - 12}{14}

3 x + 5 = \frac{π}{2} - (4 x + 1) + 2 πn

Expand

\frac{π}{2} - (4 x + 1) + 2 πn : \frac{π}{2} - 4 x - 1 + 2 πn

\frac{π}{2} - (4 x + 1) + 2 πn

- (4 x + 1) : - 4 x - 1

- (4 x + 1)

Distribute parentheses

= - (4 x) - (1)

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 1

= \frac{π}{2} - 4 x - 1 + 2 πn

3 x + 5 = \frac{π}{2} - 4 x - 1 + 2 πn

Move

5

to the right side

3 x + 5 = \frac{π}{2} - 4 x - 1 + 2 πn

Subtract

5

from both sides

3 x + 5 - 5 = \frac{π}{2} - 4 x - 1 + 2 πn - 5

Simplify

3 x + 5 - 5 = \frac{π}{2} - 4 x - 1 + 2 πn - 5

Simplify

3 x + 5 - 5 : 3 x

3 x + 5 - 5

Add similar elements:

5 - 5 = 0

= 3 x

Simplify

\frac{π}{2} - 4 x - 1 + 2 πn - 5 : - 4 x + 2 πn + \frac{π}{2} - 6

\frac{π}{2} - 4 x - 1 + 2 πn - 5

Group like terms

= - 4 x + 2 πn + \frac{π}{2} - 1 - 5

Subtract the numbers:

- 1 - 5 = - 6

= - 4 x + 2 πn + \frac{π}{2} - 6

3 x = - 4 x + 2 πn + \frac{π}{2} - 6

3 x = - 4 x + 2 πn + \frac{π}{2} - 6

3 x = - 4 x + 2 πn + \frac{π}{2} - 6

Move

4 x

to the left side

3 x = - 4 x + 2 πn + \frac{π}{2} - 6

Add

4 x

to both sides

3 x + 4 x = - 4 x + 2 πn + \frac{π}{2} - 6 + 4 x

Simplify

7 x = 2 πn + \frac{π}{2} - 6

7 x = 2 πn + \frac{π}{2} - 6

Divide both sides by

7

7 x = 2 πn + \frac{π}{2} - 6

Divide both sides by

7

\frac{7 x}{7} = \frac{2 πn}{7} + \frac{\frac{π}{2}}{7} - \frac{6}{7}

Simplify

\frac{7 x}{7} = \frac{2 πn}{7} + \frac{\frac{π}{2}}{7} - \frac{6}{7}

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{2 πn}{7} + \frac{\frac{π}{2}}{7} - \frac{6}{7} : \frac{4 πn + π - 12}{14}

\frac{2 πn}{7} + \frac{\frac{π}{2}}{7} - \frac{6}{7}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 6}{7}

Join

2 πn + \frac{π}{2} - 6 : \frac{4 πn + π - 12}{2}

2 πn + \frac{π}{2} - 6

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 6 = \frac{6 \cdot 2}{2}

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

2 πn \cdot 2 + π - 6 \cdot 2 = 4 πn + π - 12

2 πn \cdot 2 + π - 6 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 6 \cdot 2

Multiply the numbers:

6 \cdot 2 = 12

= 4 πn + π - 12

= \frac{4 πn + π - 12}{2}

= \frac{\frac{4 πn + π - 12}{2}}{7}

Apply the fraction rule:

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

= \frac{4 πn + π - 12}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{4 πn + π - 12}{14}

x = \frac{4 πn + π - 12}{14}

x = \frac{4 πn + π - 12}{14}

x = \frac{4 πn + π - 12}{14}

3 x + 5 = π - (\frac{π}{2} - (4 x + 1)) + 2 πn : x = - \frac{π + 4 πn - 8}{2}

3 x + 5 = π - (\frac{π}{2} - (4 x + 1)) + 2 πn

Expand

π - (\frac{π}{2} - (4 x + 1)) + 2 πn : π - \frac{π}{2} + 4 x + 1 + 2 πn

π - (\frac{π}{2} - (4 x + 1)) + 2 πn

- (4 x + 1) : - 4 x - 1

- (4 x + 1)

Distribute parentheses

= - (4 x) - (1)

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 1

= π - (- 4 x + \frac{π}{2} - 1) + 2 πn

- (\frac{π}{2} - 4 x - 1) : - \frac{π}{2} + 4 x + 1

- (\frac{π}{2} - 4 x - 1)

Distribute parentheses

= - (\frac{π}{2}) - (- 4 x) - (- 1)

Apply minus-plus rules

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

= - \frac{π}{2} + 4 x + 1

= π - \frac{π}{2} + 4 x + 1 + 2 πn

3 x + 5 = π - \frac{π}{2} + 4 x + 1 + 2 πn

Move

5

to the right side

3 x + 5 = π - \frac{π}{2} + 4 x + 1 + 2 πn

Subtract

5

from both sides

3 x + 5 - 5 = π - \frac{π}{2} + 4 x + 1 + 2 πn - 5

Simplify

3 x + 5 - 5 = π - \frac{π}{2} + 4 x + 1 + 2 πn - 5

Simplify

3 x + 5 - 5 : 3 x

3 x + 5 - 5

Add similar elements:

5 - 5 = 0

= 3 x

Simplify

π - \frac{π}{2} + 4 x + 1 + 2 πn - 5 : 4 x + 2 πn + π - 4 - \frac{π}{2}

π - \frac{π}{2} + 4 x + 1 + 2 πn - 5

Group like terms

= 4 x + π + 2 πn - \frac{π}{2} + 1 - 5

Add/Subtract the numbers:

1 - 5 = - 4

= 4 x + 2 πn + π - 4 - \frac{π}{2}

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

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

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

Move

4 x

to the left side

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

Subtract

4 x

from both sides

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

Simplify

- x = 2 πn + π - 4 - \frac{π}{2}

- x = 2 πn + π - 4 - \frac{π}{2}

Divide both sides by

- 1

- x = 2 πn + π - 4 - \frac{π}{2}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{4}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{4}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{4}{- 1} - \frac{\frac{π}{2}}{- 1} : - \frac{π + 4 πn - 8}{2}

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{4}{- 1} - \frac{\frac{π}{2}}{- 1}

Apply rule

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

= \frac{2 πn + π - 4 - \frac{π}{2}}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 πn + π - 4 - \frac{π}{2}}{1}

Join

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

2 πn + π - 4 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 4 = \frac{4 \cdot 2}{2}

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

2 πn \cdot 2 + π 2 - 4 \cdot 2 - π = π + 4 πn - 8

2 πn \cdot 2 + π 2 - 4 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 4 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 4 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 4 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= π + 4 πn - 8

= \frac{π + 4 πn - 8}{2}

= - \frac{\frac{π + 4 πn - 8}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn - 8}{2}

x = - \frac{π + 4 πn - 8}{2}

x = - \frac{π + 4 πn - 8}{2}

x = - \frac{π + 4 πn - 8}{2}

x = \frac{4 πn + π - 12}{14}, x = - \frac{π + 4 πn - 8}{2}

x = \frac{4 πn + π - 12}{14}, x = - \frac{π + 4 πn - 8}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x+5)=cos(4x+1) ?
The general solution for sin(3x+5)=cos(4x+1) is x=(4pin+pi-12}{14},x=-\frac{pi+4pin-8)/2