What is the general solution for cos(10x)=sin(5x-3) ?

The general solution for cos(10x)=sin(5x-3) is x=(6+4pin+pi)/(30),x=-(pi+4pin+6)/(10)

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cos(10x)=sin(5x-3)

Solution

cos (10 x) = sin (5 x - 3)

Solution

x = \frac{6 + 4 πn + π}{30}, x = - \frac{π + 4 πn + 6}{10}

Degrees

x = 17.45915 \dots^{\circ} + 2 4^{\circ} n, x = - 52.37746 \dots^{\circ} - 7 2^{\circ} n

Solution steps

cos (10 x) = sin (5 x - 3)

Rewrite using trig identities

cos (10 x) = sin (5 x - 3)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (10 x) = sin (\frac{π}{2} - 10 x)

cos (10 x) = sin (\frac{π}{2} - 10 x)

Apply trig inverse properties

cos (10 x) = sin (\frac{π}{2} - 10 x)

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

5 x - 3 = \frac{π}{2} - 10 x + 2 πn, 5 x - 3 = π - (\frac{π}{2} - 10 x) + 2 πn

5 x - 3 = \frac{π}{2} - 10 x + 2 πn, 5 x - 3 = π - (\frac{π}{2} - 10 x) + 2 πn

5 x - 3 = \frac{π}{2} - 10 x + 2 πn : x = \frac{6 + 4 πn + π}{30}

5 x - 3 = \frac{π}{2} - 10 x + 2 πn

Move

3

to the right side

5 x - 3 = \frac{π}{2} - 10 x + 2 πn

Add

3

to both sides

5 x - 3 + 3 = \frac{π}{2} - 10 x + 2 πn + 3

Simplify

5 x = \frac{π}{2} - 10 x + 2 πn + 3

5 x = \frac{π}{2} - 10 x + 2 πn + 3

Move

10 x

to the left side

5 x = \frac{π}{2} - 10 x + 2 πn + 3

Add

10 x

to both sides

5 x + 10 x = \frac{π}{2} - 10 x + 2 πn + 3 + 10 x

Simplify

15 x = \frac{π}{2} + 2 πn + 3

15 x = \frac{π}{2} + 2 πn + 3

Divide both sides by

15

15 x = \frac{π}{2} + 2 πn + 3

Divide both sides by

15

\frac{15 x}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15} + \frac{3}{15}

Simplify

\frac{15 x}{15} = \frac{\frac{π}{2}}{15} + \frac{2 πn}{15} + \frac{3}{15}

Simplify

\frac{15 x}{15} : x

\frac{15 x}{15}

Divide the numbers:

\frac{15}{15} = 1

= x

Simplify

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15} + \frac{3}{15} : \frac{6 + 4 πn + π}{30}

\frac{\frac{π}{2}}{15} + \frac{2 πn}{15} + \frac{3}{15}

Group like terms

= \frac{3}{15} + \frac{2 πn}{15} + \frac{\frac{π}{2}}{15}

Apply rule

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

= \frac{3 + 2 πn + \frac{π}{2}}{15}

Join

3 + 2 πn + \frac{π}{2} : \frac{6 + 4 πn + π}{2}

3 + 2 πn + \frac{π}{2}

Convert element to fraction:

3 = \frac{3 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{3 \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{3 \cdot 2 + 2 πn \cdot 2 + π}{2}

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

3 \cdot 2 + 2 πn \cdot 2 + π

Multiply the numbers:

3 \cdot 2 = 6

= 6 + 2 \cdot 2 πn + π

Multiply the numbers:

2 \cdot 2 = 4

= 6 + 4 πn + π

= \frac{6 + 4 πn + π}{2}

= \frac{\frac{6 + 4 πn + π}{2}}{15}

Apply the fraction rule:

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

= \frac{6 + 4 πn + π}{2 \cdot 15}

Multiply the numbers:

2 \cdot 15 = 30

= \frac{6 + 4 πn + π}{30}

x = \frac{6 + 4 πn + π}{30}

x = \frac{6 + 4 πn + π}{30}

x = \frac{6 + 4 πn + π}{30}

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

5 x - 3 = π - (\frac{π}{2} - 10 x) + 2 πn

Expand

π - (\frac{π}{2} - 10 x) + 2 πn : π - \frac{π}{2} + 10 x + 2 πn

π - (\frac{π}{2} - 10 x) + 2 πn

- (\frac{π}{2} - 10 x) : - \frac{π}{2} + 10 x

- (\frac{π}{2} - 10 x)

Distribute parentheses

= - (\frac{π}{2}) - (- 10 x)

Apply minus-plus rules

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

= - \frac{π}{2} + 10 x

= π - \frac{π}{2} + 10 x + 2 πn

5 x - 3 = π - \frac{π}{2} + 10 x + 2 πn

Move

3

to the right side

5 x - 3 = π - \frac{π}{2} + 10 x + 2 πn

Add

3

to both sides

5 x - 3 + 3 = π - \frac{π}{2} + 10 x + 2 πn + 3

Simplify

5 x = π - \frac{π}{2} + 10 x + 2 πn + 3

5 x = π - \frac{π}{2} + 10 x + 2 πn + 3

Move

10 x

to the left side

5 x = π - \frac{π}{2} + 10 x + 2 πn + 3

Subtract

10 x

from both sides

5 x - 10 x = π - \frac{π}{2} + 10 x + 2 πn + 3 - 10 x

Simplify

- 5 x = π - \frac{π}{2} + 2 πn + 3

- 5 x = π - \frac{π}{2} + 2 πn + 3

Divide both sides by

- 5

- 5 x = π - \frac{π}{2} + 2 πn + 3

Divide both sides by

- 5

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

Simplify

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

Simplify

\frac{- 5 x}{- 5} : x

\frac{- 5 x}{- 5}

Apply the fraction rule:

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

= \frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{π}{- 5} - \frac{\frac{π}{2}}{- 5} + \frac{2 πn}{- 5} + \frac{3}{- 5} : - \frac{π + 4 πn + 6}{10}

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

Group like terms

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

Apply rule

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

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

Apply the fraction rule:

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

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

Join

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

π + 3 + 2 πn - \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}, 3 = \frac{3 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} + \frac{3 \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 + 3 \cdot 2 + 2 πn \cdot 2 - π}{2}

π 2 + 3 \cdot 2 + 2 πn \cdot 2 - π = π + 4 πn + 6

π 2 + 3 \cdot 2 + 2 πn \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn + 3 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn + 3 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn + 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= π + 4 πn + 6

= \frac{π + 4 πn + 6}{2}

= - \frac{\frac{π + 4 πn + 6}{2}}{5}

Simplify

\frac{\frac{π + 4 πn + 6}{2}}{5} : \frac{π + 4 πn + 6}{10}

\frac{\frac{π + 4 πn + 6}{2}}{5}

Apply the fraction rule:

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

= \frac{π + 4 πn + 6}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{π + 4 πn + 6}{10}

= - \frac{π + 4 πn + 6}{10}

x = - \frac{π + 4 πn + 6}{10}

x = - \frac{π + 4 πn + 6}{10}

x = - \frac{π + 4 πn + 6}{10}

x = \frac{6 + 4 πn + π}{30}, x = - \frac{π + 4 πn + 6}{10}

x = \frac{6 + 4 πn + π}{30}, x = - \frac{π + 4 πn + 6}{10}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(10x)=sin(5x-3) ?
The general solution for cos(10x)=sin(5x-3) is x=(6+4pin+pi)/(30),x=-(pi+4pin+6)/(10)