What is the general solution for cos(2x-1)=sin(3x+6) ?

The general solution for cos(2x-1)=sin(3x+6) is x=(4pin+pi-10)/(10),x=2pin+pi-7-pi/2

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cos(2x-1)=sin(3x+6)

Solution

cos (2 x - 1) = sin (3 x + 6)

Solution

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

Degrees

x = - 39.29577 \dots^{\circ} + 7 2^{\circ} n, x = - 311.07045 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos (2 x - 1) = sin (3 x + 6)

Rewrite using trig identities

cos (2 x - 1) = sin (3 x + 6)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

Apply trig inverse properties

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

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

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn, 3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn, 3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn : x = \frac{4 πn + π - 10}{10}

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn

Expand

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

\frac{π}{2} - (2 x - 1) + 2 πn

- (2 x - 1) : - 2 x + 1

- (2 x - 1)

Distribute parentheses

= - (2 x) - (- 1)

Apply minus-plus rules

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

= - 2 x + 1

= \frac{π}{2} - 2 x + 1 + 2 πn

3 x + 6 = \frac{π}{2} - 2 x + 1 + 2 πn

Move

6

to the right side

3 x + 6 = \frac{π}{2} - 2 x + 1 + 2 πn

Subtract

6

from both sides

3 x + 6 - 6 = \frac{π}{2} - 2 x + 1 + 2 πn - 6

Simplify

3 x + 6 - 6 = \frac{π}{2} - 2 x + 1 + 2 πn - 6

Simplify

3 x + 6 - 6 : 3 x

3 x + 6 - 6

Add similar elements:

6 - 6 = 0

= 3 x

Simplify

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

\frac{π}{2} - 2 x + 1 + 2 πn - 6

Group like terms

= - 2 x + 2 πn + \frac{π}{2} + 1 - 6

Add/Subtract the numbers:

1 - 6 = - 5

= - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

Move

2 x

to the left side

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

Add

2 x

to both sides

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

Simplify

5 x = 2 πn + \frac{π}{2} - 5

5 x = 2 πn + \frac{π}{2} - 5

Divide both sides by

5

5 x = 2 πn + \frac{π}{2} - 5

Divide both sides by

5

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

Simplify

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

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{2 πn}{5} + \frac{\frac{π}{2}}{5} - \frac{5}{5} : \frac{4 πn + π - 10}{10}

\frac{2 πn}{5} + \frac{\frac{π}{2}}{5} - \frac{5}{5}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 5}{5}

Join

2 πn + \frac{π}{2} - 5 : \frac{4 πn + π - 10}{2}

2 πn + \frac{π}{2} - 5

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 5 = \frac{5 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{5 \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 - 5 \cdot 2}{2}

2 πn \cdot 2 + π - 5 \cdot 2 = 4 πn + π - 10

2 πn \cdot 2 + π - 5 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 4 πn + π - 10

= \frac{4 πn + π - 10}{2}

= \frac{\frac{4 πn + π - 10}{2}}{5}

Apply the fraction rule:

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

= \frac{4 πn + π - 10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn : x = 2 πn + π - 7 - \frac{π}{2}

3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

Expand

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

π - (\frac{π}{2} - (2 x - 1)) + 2 πn

- (2 x - 1) : - 2 x + 1

- (2 x - 1)

Distribute parentheses

= - (2 x) - (- 1)

Apply minus-plus rules

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

= - 2 x + 1

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

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

- (\frac{π}{2} - 2 x + 1)

Distribute parentheses

= - (\frac{π}{2}) - (- 2 x) - (1)

Apply minus-plus rules

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

= - \frac{π}{2} + 2 x - 1

= π - \frac{π}{2} + 2 x - 1 + 2 πn

3 x + 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn

Move

6

to the right side

3 x + 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn

Subtract

6

from both sides

3 x + 6 - 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

Simplify

3 x + 6 - 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

Simplify

3 x + 6 - 6 : 3 x

3 x + 6 - 6

Add similar elements:

6 - 6 = 0

= 3 x

Simplify

π - \frac{π}{2} + 2 x - 1 + 2 πn - 6 : 2 x + 2 πn + π - 7 - \frac{π}{2}

π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

Group like terms

= 2 x + π + 2 πn - \frac{π}{2} - 1 - 6

Subtract the numbers:

- 1 - 6 = - 7

= 2 x + 2 πn + π - 7 - \frac{π}{2}

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

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

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

Move

2 x

to the left side

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

Subtract

2 x

from both sides

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

Simplify

x = 2 πn + π - 7 - \frac{π}{2}

x = 2 πn + π - 7 - \frac{π}{2}

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x-1)=sin(3x+6) ?
The general solution for cos(2x-1)=sin(3x+6) is x=(4pin+pi-10)/(10),x=2pin+pi-7-pi/2