What is the general solution for sin(2x+7)=cos(4x-7) ?

The general solution for sin(2x+7)=cos(4x-7) is x=(4pin+pi}{12},x=-\frac{pi+4pin-28)/4

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sin(2x+7)=cos(4x-7)

Solution

sin (2 x + 7) = cos (4 x - 7)

Solution

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

Degrees

x = 1 5^{\circ} + 6 0^{\circ} n, x = 356.07045 \dots^{\circ} - 18 0^{\circ} n

Solution steps

sin (2 x + 7) = cos (4 x - 7)

Rewrite using trig identities

sin (2 x + 7) = cos (4 x - 7)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (2 x + 7) = sin (\frac{π}{2} - (4 x - 7))

sin (2 x + 7) = sin (\frac{π}{2} - (4 x - 7))

Apply trig inverse properties

sin (2 x + 7) = sin (\frac{π}{2} - (4 x - 7))

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

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

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

2 x + 7 = \frac{π}{2} - (4 x - 7) + 2 πn : x = \frac{4 πn + π}{12}

2 x + 7 = \frac{π}{2} - (4 x - 7) + 2 πn

Expand

\frac{π}{2} - (4 x - 7) + 2 πn : \frac{π}{2} - 4 x + 7 + 2 πn

\frac{π}{2} - (4 x - 7) + 2 πn

- (4 x - 7) : - 4 x + 7

- (4 x - 7)

Distribute parentheses

= - (4 x) - (- 7)

Apply minus-plus rules

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

= - 4 x + 7

= \frac{π}{2} - 4 x + 7 + 2 πn

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

Move

7

to the right side

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

Subtract

7

from both sides

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

Simplify

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

Simplify

2 x + 7 - 7 : 2 x

2 x + 7 - 7

Add similar elements:

7 - 7 = 0

= 2 x

Simplify

\frac{π}{2} - 4 x + 7 + 2 πn - 7 : - 4 x + 2 πn + \frac{π}{2}

\frac{π}{2} - 4 x + 7 + 2 πn - 7

Group like terms

= - 4 x + 2 πn + \frac{π}{2} + 7 - 7

7 - 7 = 0

= - 4 x + 2 πn + \frac{π}{2}

2 x = - 4 x + 2 πn + \frac{π}{2}

2 x = - 4 x + 2 πn + \frac{π}{2}

2 x = - 4 x + 2 πn + \frac{π}{2}

Move

4 x

to the left side

2 x = - 4 x + 2 πn + \frac{π}{2}

Add

4 x

to both sides

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

Simplify

6 x = 2 πn + \frac{π}{2}

6 x = 2 πn + \frac{π}{2}

Divide both sides by

6

6 x = 2 πn + \frac{π}{2}

Divide both sides by

6

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

Simplify

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

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

Apply rule

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

= \frac{2 πn + \frac{π}{2}}{6}

Join

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

2 πn + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{2 πn \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}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 πn + π}{2}

= \frac{\frac{4 πn + π}{2}}{6}

Apply the fraction rule:

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

= \frac{4 πn + π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4 πn + π}{12}

x = \frac{4 πn + π}{12}

x = \frac{4 πn + π}{12}

x = \frac{4 πn + π}{12}

2 x + 7 = π - (\frac{π}{2} - (4 x - 7)) + 2 πn : x = - \frac{π + 4 πn - 28}{4}

2 x + 7 = π - (\frac{π}{2} - (4 x - 7)) + 2 πn

Expand

π - (\frac{π}{2} - (4 x - 7)) + 2 πn : π - \frac{π}{2} + 4 x - 7 + 2 πn

π - (\frac{π}{2} - (4 x - 7)) + 2 πn

- (4 x - 7) : - 4 x + 7

- (4 x - 7)

Distribute parentheses

= - (4 x) - (- 7)

Apply minus-plus rules

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

= - 4 x + 7

= π - (- 4 x + 7 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 4 x + 7) : - \frac{π}{2} + 4 x - 7

- (\frac{π}{2} - 4 x + 7)

Distribute parentheses

= - (\frac{π}{2}) - (- 4 x) - (7)

Apply minus-plus rules

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

= - \frac{π}{2} + 4 x - 7

= π - \frac{π}{2} + 4 x - 7 + 2 πn

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

Move

7

to the right side

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

Subtract

7

from both sides

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

Simplify

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

Simplify

2 x + 7 - 7 : 2 x

2 x + 7 - 7

Add similar elements:

7 - 7 = 0

= 2 x

Simplify

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

π - \frac{π}{2} + 4 x - 7 + 2 πn - 7

Group like terms

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

Subtract the numbers:

- 7 - 7 = - 14

= 4 x + 2 πn + π - 14 - \frac{π}{2}

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

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

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

Move

4 x

to the left side

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

Subtract

4 x

from both sides

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

Simplify

- 2 x = 2 πn + π - 14 - \frac{π}{2}

- 2 x = 2 πn + π - 14 - \frac{π}{2}

Divide both sides by

- 2

- 2 x = 2 πn + π - 14 - \frac{π}{2}

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{- 2} + \frac{π}{- 2} - \frac{14}{- 2} - \frac{\frac{π}{2}}{- 2} : - \frac{π + 4 πn - 28}{4}

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

Apply rule

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

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

Apply the fraction rule:

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

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

Join

2 πn + π - 14 - \frac{π}{2} : \frac{π + 4 πn - 28}{2}

2 πn + π - 14 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 14 = \frac{14 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{14 \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 - 14 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 14 \cdot 2 - π = π + 4 πn - 28

2 πn \cdot 2 + π 2 - 14 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 14 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 14 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 14 \cdot 2

Multiply the numbers:

14 \cdot 2 = 28

= π + 4 πn - 28

= \frac{π + 4 πn - 28}{2}

= - \frac{\frac{π + 4 πn - 28}{2}}{2}

Simplify

\frac{\frac{π + 4 πn - 28}{2}}{2} : \frac{π + 4 πn - 28}{4}

\frac{\frac{π + 4 πn - 28}{2}}{2}

Apply the fraction rule:

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

= \frac{π + 4 πn - 28}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn - 28}{4}

= - \frac{π + 4 πn - 28}{4}

x = - \frac{π + 4 πn - 28}{4}

x = - \frac{π + 4 πn - 28}{4}

x = - \frac{π + 4 πn - 28}{4}

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

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+7)=cos(4x-7) ?
The general solution for sin(2x+7)=cos(4x-7) is x=(4pin+pi}{12},x=-\frac{pi+4pin-28)/4