What is the general solution for sin(2x+10)=cos(3x-20) ?

The general solution for sin(2x+10)=cos(3x-20) is x=(4pin+20+pi}{10},x=-\frac{pi+4pin-60)/2

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sin(2x+10)=cos(3x-20)

Solution

sin (2 x + 10) = cos (3 x - 20)

Solution

x = \frac{4 πn + 20 + π}{10}, x = - \frac{π + 4 πn - 60}{2}

Degrees

x = 132.59155 \dots^{\circ} + 7 2^{\circ} n, x = 1628.87338 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (2 x + 10) = cos (3 x - 20)

Rewrite using trig identities

sin (2 x + 10) = cos (3 x - 20)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (2 x + 10) = sin (\frac{π}{2} - (3 x - 20))

sin (2 x + 10) = sin (\frac{π}{2} - (3 x - 20))

Apply trig inverse properties

sin (2 x + 10) = sin (\frac{π}{2} - (3 x - 20))

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

2 x + 10 = \frac{π}{2} - (3 x - 20) + 2 πn, 2 x + 10 = π - (\frac{π}{2} - (3 x - 20)) + 2 πn

2 x + 10 = \frac{π}{2} - (3 x - 20) + 2 πn, 2 x + 10 = π - (\frac{π}{2} - (3 x - 20)) + 2 πn

2 x + 10 = \frac{π}{2} - (3 x - 20) + 2 πn : x = \frac{4 πn + 20 + π}{10}

2 x + 10 = \frac{π}{2} - (3 x - 20) + 2 πn

Expand

\frac{π}{2} - (3 x - 20) + 2 πn : \frac{π}{2} - 3 x + 20 + 2 πn

\frac{π}{2} - (3 x - 20) + 2 πn

- (3 x - 20) : - 3 x + 20

- (3 x - 20)

Distribute parentheses

= - (3 x) - (- 20)

Apply minus-plus rules

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

= - 3 x + 20

= \frac{π}{2} - 3 x + 20 + 2 πn

2 x + 10 = \frac{π}{2} - 3 x + 20 + 2 πn

Move

10

to the right side

2 x + 10 = \frac{π}{2} - 3 x + 20 + 2 πn

Subtract

10

from both sides

2 x + 10 - 10 = \frac{π}{2} - 3 x + 20 + 2 πn - 10

Simplify

2 x + 10 - 10 = \frac{π}{2} - 3 x + 20 + 2 πn - 10

Simplify

2 x + 10 - 10 : 2 x

2 x + 10 - 10

Add similar elements:

10 - 10 = 0

= 2 x

Simplify

\frac{π}{2} - 3 x + 20 + 2 πn - 10 : - 3 x + 2 πn + 10 + \frac{π}{2}

\frac{π}{2} - 3 x + 20 + 2 πn - 10

Group like terms

= - 3 x + 2 πn + \frac{π}{2} + 20 - 10

Add/Subtract the numbers:

20 - 10 = 10

= - 3 x + 2 πn + 10 + \frac{π}{2}

2 x = - 3 x + 2 πn + 10 + \frac{π}{2}

2 x = - 3 x + 2 πn + 10 + \frac{π}{2}

2 x = - 3 x + 2 πn + 10 + \frac{π}{2}

Move

3 x

to the left side

2 x = - 3 x + 2 πn + 10 + \frac{π}{2}

Add

3 x

to both sides

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

Simplify

5 x = 2 πn + 10 + \frac{π}{2}

5 x = 2 πn + 10 + \frac{π}{2}

Divide both sides by

5

5 x = 2 πn + 10 + \frac{π}{2}

Divide both sides by

5

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

Simplify

\frac{5 x}{5} = \frac{2 πn}{5} + \frac{10}{5} + \frac{\frac{π}{2}}{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{10}{5} + \frac{\frac{π}{2}}{5} : \frac{4 πn + 20 + π}{10}

\frac{2 πn}{5} + \frac{10}{5} + \frac{\frac{π}{2}}{5}

Apply rule

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

= \frac{2 πn + 10 + \frac{π}{2}}{5}

Join

2 πn + 10 + \frac{π}{2} : \frac{4 πn + 20 + π}{2}

2 πn + 10 + \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 10 = \frac{10 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{10 \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 + 10 \cdot 2 + π}{2}

2 πn \cdot 2 + 10 \cdot 2 + π = 4 πn + 20 + π

2 πn \cdot 2 + 10 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 10 \cdot 2 + π

Multiply the numbers:

10 \cdot 2 = 20

= 4 πn + 20 + π

= \frac{4 πn + 20 + π}{2}

= \frac{\frac{4 πn + 20 + π}{2}}{5}

Apply the fraction rule:

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

= \frac{4 πn + 20 + π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{4 πn + 20 + π}{10}

x = \frac{4 πn + 20 + π}{10}

x = \frac{4 πn + 20 + π}{10}

x = \frac{4 πn + 20 + π}{10}

2 x + 10 = π - (\frac{π}{2} - (3 x - 20)) + 2 πn : x = - \frac{π + 4 πn - 60}{2}

2 x + 10 = π - (\frac{π}{2} - (3 x - 20)) + 2 πn

Expand

π - (\frac{π}{2} - (3 x - 20)) + 2 πn : π - \frac{π}{2} + 3 x - 20 + 2 πn

π - (\frac{π}{2} - (3 x - 20)) + 2 πn

- (3 x - 20) : - 3 x + 20

- (3 x - 20)

Distribute parentheses

= - (3 x) - (- 20)

Apply minus-plus rules

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

= - 3 x + 20

= π - (- 3 x + 20 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 3 x + 20) : - \frac{π}{2} + 3 x - 20

- (\frac{π}{2} - 3 x + 20)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x) - (20)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x - 20

= π - \frac{π}{2} + 3 x - 20 + 2 πn

2 x + 10 = π - \frac{π}{2} + 3 x - 20 + 2 πn

Move

10

to the right side

2 x + 10 = π - \frac{π}{2} + 3 x - 20 + 2 πn

Subtract

10

from both sides

2 x + 10 - 10 = π - \frac{π}{2} + 3 x - 20 + 2 πn - 10

Simplify

2 x + 10 - 10 = π - \frac{π}{2} + 3 x - 20 + 2 πn - 10

Simplify

2 x + 10 - 10 : 2 x

2 x + 10 - 10

Add similar elements:

10 - 10 = 0

= 2 x

Simplify

π - \frac{π}{2} + 3 x - 20 + 2 πn - 10 : 3 x + 2 πn + π - 30 - \frac{π}{2}

π - \frac{π}{2} + 3 x - 20 + 2 πn - 10

Group like terms

= 3 x + π + 2 πn - \frac{π}{2} - 20 - 10

Subtract the numbers:

- 20 - 10 = - 30

= 3 x + 2 πn + π - 30 - \frac{π}{2}

2 x = 3 x + 2 πn + π - 30 - \frac{π}{2}

2 x = 3 x + 2 πn + π - 30 - \frac{π}{2}

2 x = 3 x + 2 πn + π - 30 - \frac{π}{2}

Move

3 x

to the left side

2 x = 3 x + 2 πn + π - 30 - \frac{π}{2}

Subtract

3 x

from both sides

2 x - 3 x = 3 x + 2 πn + π - 30 - \frac{π}{2} - 3 x

Simplify

- x = 2 πn + π - 30 - \frac{π}{2}

- x = 2 πn + π - 30 - \frac{π}{2}

Divide both sides by

- 1

- x = 2 πn + π - 30 - \frac{π}{2}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{30}{- 1} - \frac{\frac{π}{2}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{30}{- 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{30}{- 1} - \frac{\frac{π}{2}}{- 1} : - \frac{π + 4 πn - 60}{2}

\frac{2 πn}{- 1} + \frac{π}{- 1} - \frac{30}{- 1} - \frac{\frac{π}{2}}{- 1}

Apply rule

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

= \frac{2 πn + π - 30 - \frac{π}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn + π - 30 - \frac{π}{2}}{1}

Join

2 πn + π - 30 - \frac{π}{2} : \frac{π + 4 πn - 60}{2}

2 πn + π - 30 - \frac{π}{2}

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 30 = \frac{30 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{30 \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 - 30 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 30 \cdot 2 - π = π + 4 πn - 60

2 πn \cdot 2 + π 2 - 30 \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn - 30 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn - 30 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn - 30 \cdot 2

Multiply the numbers:

30 \cdot 2 = 60

= π + 4 πn - 60

= \frac{π + 4 πn - 60}{2}

= - \frac{\frac{π + 4 πn - 60}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn - 60}{2}

x = - \frac{π + 4 πn - 60}{2}

x = - \frac{π + 4 πn - 60}{2}

x = - \frac{π + 4 πn - 60}{2}

x = \frac{4 πn + 20 + π}{10}, x = - \frac{π + 4 πn - 60}{2}

x = \frac{4 πn + 20 + π}{10}, x = - \frac{π + 4 πn - 60}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+10)=cos(3x-20) ?
The general solution for sin(2x+10)=cos(3x-20) is x=(4pin+20+pi}{10},x=-\frac{pi+4pin-60)/2