What is the general solution for cos(7x)=sin(5x-6) ?

The general solution for cos(7x)=sin(5x-6) is x=(12+4pin+pi}{24},x=-\frac{pi+4pin+12)/4

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cos(7x)=sin(5x-6)

Solution

cos (7 x) = sin (5 x - 6)

Solution

x = \frac{12 + 4 πn + π}{24}, x = - \frac{π + 4 πn + 12}{4}

Degrees

x = 36.14788 \dots^{\circ} + 3 0^{\circ} n, x = - 216.88733 \dots^{\circ} - 18 0^{\circ} n

Solution steps

cos (7 x) = sin (5 x - 6)

Rewrite using trig identities

cos (7 x) = sin (5 x - 6)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (7 x) = sin (\frac{π}{2} - 7 x)

cos (7 x) = sin (\frac{π}{2} - 7 x)

Apply trig inverse properties

cos (7 x) = sin (\frac{π}{2} - 7 x)

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

5 x - 6 = \frac{π}{2} - 7 x + 2 πn, 5 x - 6 = π - (\frac{π}{2} - 7 x) + 2 πn

5 x - 6 = \frac{π}{2} - 7 x + 2 πn, 5 x - 6 = π - (\frac{π}{2} - 7 x) + 2 πn

5 x - 6 = \frac{π}{2} - 7 x + 2 πn : x = \frac{12 + 4 πn + π}{24}

5 x - 6 = \frac{π}{2} - 7 x + 2 πn

Move

6

to the right side

5 x - 6 = \frac{π}{2} - 7 x + 2 πn

Add

6

to both sides

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

Simplify

5 x = \frac{π}{2} - 7 x + 2 πn + 6

5 x = \frac{π}{2} - 7 x + 2 πn + 6

Move

7 x

to the left side

5 x = \frac{π}{2} - 7 x + 2 πn + 6

Add

7 x

to both sides

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

Simplify

12 x = \frac{π}{2} + 2 πn + 6

12 x = \frac{π}{2} + 2 πn + 6

Divide both sides by

12

12 x = \frac{π}{2} + 2 πn + 6

Divide both sides by

12

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

Simplify

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

Simplify

\frac{12 x}{12} : x

\frac{12 x}{12}

Divide the numbers:

\frac{12}{12} = 1

= x

Simplify

\frac{\frac{π}{2}}{12} + \frac{2 πn}{12} + \frac{6}{12} : \frac{12 + 4 πn + π}{24}

\frac{\frac{π}{2}}{12} + \frac{2 πn}{12} + \frac{6}{12}

Group like terms

= \frac{6}{12} + \frac{2 πn}{12} + \frac{\frac{π}{2}}{12}

Apply rule

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

= \frac{6 + 2 πn + \frac{π}{2}}{12}

Join

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

6 + 2 πn + \frac{π}{2}

Convert element to fraction:

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

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

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

6 \cdot 2 + 2 πn \cdot 2 + π

Multiply the numbers:

6 \cdot 2 = 12

= 12 + 2 \cdot 2 πn + π

Multiply the numbers:

2 \cdot 2 = 4

= 12 + 4 πn + π

= \frac{12 + 4 πn + π}{2}

= \frac{\frac{12 + 4 πn + π}{2}}{12}

Apply the fraction rule:

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

= \frac{12 + 4 πn + π}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{12 + 4 πn + π}{24}

x = \frac{12 + 4 πn + π}{24}

x = \frac{12 + 4 πn + π}{24}

x = \frac{12 + 4 πn + π}{24}

5 x - 6 = π - (\frac{π}{2} - 7 x) + 2 πn : x = - \frac{π + 4 πn + 12}{4}

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

Expand

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

π - (\frac{π}{2} - 7 x) + 2 πn

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

- (\frac{π}{2} - 7 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 7 x)

Apply minus-plus rules

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

= - \frac{π}{2} + 7 x

= π - \frac{π}{2} + 7 x + 2 πn

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

Move

6

to the right side

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

Add

6

to both sides

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

Simplify

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

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

Move

7 x

to the left side

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

Subtract

7 x

from both sides

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

Simplify

- 2 x = π - \frac{π}{2} + 2 πn + 6

- 2 x = π - \frac{π}{2} + 2 πn + 6

Divide both sides by

- 2

- 2 x = π - \frac{π}{2} + 2 πn + 6

Divide both sides by

- 2

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

Simplify

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

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

Group like terms

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

Apply rule

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

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

Apply the fraction rule:

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

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

Join

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

π + 6 + 2 πn - \frac{π}{2}

Convert element to fraction:

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

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

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

π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn + 6 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn + 6 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn + 6 \cdot 2

Multiply the numbers:

6 \cdot 2 = 12

= π + 4 πn + 12

= \frac{π + 4 πn + 12}{2}

= - \frac{\frac{π + 4 πn + 12}{2}}{2}

Simplify

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

\frac{\frac{π + 4 πn + 12}{2}}{2}

Apply the fraction rule:

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

= \frac{π + 4 πn + 12}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn + 12}{4}

= - \frac{π + 4 πn + 12}{4}

x = - \frac{π + 4 πn + 12}{4}

x = - \frac{π + 4 πn + 12}{4}

x = - \frac{π + 4 πn + 12}{4}

x = \frac{12 + 4 πn + π}{24}, x = - \frac{π + 4 πn + 12}{4}

x = \frac{12 + 4 πn + π}{24}, x = - \frac{π + 4 πn + 12}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(7x)=sin(5x-6) ?
The general solution for cos(7x)=sin(5x-6) is x=(12+4pin+pi}{24},x=-\frac{pi+4pin+12)/4