What is the general solution for sin(2x-20)=-cos(3x+50) ?

The general solution for sin(2x-20)=-cos(3x+50) is x=2pin+pi/2-70,x=-(4pin+60+pi)/(10)

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Popular Trigonometry >

sin(2x-20)=-cos(3x+50)

Solution

sin (2 x - 20) = - cos (3 x + 50)

Solution

x = 2 πn + \frac{π}{2} - 70, x = - \frac{4 πn + 60 + π}{10}

Degrees

x = - 3920.70456 \dots^{\circ} + 36 0^{\circ} n, x = - 361.77467 \dots^{\circ} - 7 2^{\circ} n

Solution steps

sin (2 x - 20) = - cos (3 x + 50)

Multiply by

- 1

- sin (2 x - 20) = cos (3 x + 50)

Rewrite using trig identities

- sin (2 x - 20) = cos (3 x + 50)

Use the following identity:

- sin (x) = sin (- x)

sin (- (2 x - 20)) = cos (3 x + 50)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (- (2 x - 20)) = sin (\frac{π}{2} - (3 x + 50))

sin (- (2 x - 20)) = sin (\frac{π}{2} - (3 x + 50))

Apply trig inverse properties

sin (- (2 x - 20)) = sin (\frac{π}{2} - (3 x + 50))

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

- (2 x - 20) = \frac{π}{2} - (3 x + 50) + 2 πn, - (2 x - 20) = π - (\frac{π}{2} - (3 x + 50)) + 2 πn

- (2 x - 20) = \frac{π}{2} - (3 x + 50) + 2 πn, - (2 x - 20) = π - (\frac{π}{2} - (3 x + 50)) + 2 πn

- (2 x - 20) = \frac{π}{2} - (3 x + 50) + 2 πn : x = 2 πn + \frac{π}{2} - 70

- (2 x - 20) = \frac{π}{2} - (3 x + 50) + 2 πn

Expand

- (2 x - 20) : - 2 x + 20

- (2 x - 20)

Distribute parentheses

= - (2 x) - (- 20)

Apply minus-plus rules

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

= - 2 x + 20

Expand

\frac{π}{2} - (3 x + 50) + 2 πn : \frac{π}{2} - 3 x - 50 + 2 πn

\frac{π}{2} - (3 x + 50) + 2 πn

- (3 x + 50) : - 3 x - 50

- (3 x + 50)

Distribute parentheses

= - (3 x) - (50)

Apply minus-plus rules

+ (- a) = - a

= - 3 x - 50

= \frac{π}{2} - 3 x - 50 + 2 πn

- 2 x + 20 = \frac{π}{2} - 3 x - 50 + 2 πn

Move

20

to the right side

- 2 x + 20 = \frac{π}{2} - 3 x - 50 + 2 πn

Subtract

20

from both sides

- 2 x + 20 - 20 = \frac{π}{2} - 3 x - 50 + 2 πn - 20

Simplify

- 2 x + 20 - 20 = \frac{π}{2} - 3 x - 50 + 2 πn - 20

Simplify

- 2 x + 20 - 20 : - 2 x

- 2 x + 20 - 20

Add similar elements:

20 - 20 = 0

= - 2 x

Simplify

\frac{π}{2} - 3 x - 50 + 2 πn - 20 : - 3 x + 2 πn + \frac{π}{2} - 70

\frac{π}{2} - 3 x - 50 + 2 πn - 20

Group like terms

= - 3 x + 2 πn + \frac{π}{2} - 50 - 20

Subtract the numbers:

- 50 - 20 = - 70

= - 3 x + 2 πn + \frac{π}{2} - 70

- 2 x = - 3 x + 2 πn + \frac{π}{2} - 70

- 2 x = - 3 x + 2 πn + \frac{π}{2} - 70

- 2 x = - 3 x + 2 πn + \frac{π}{2} - 70

Move

3 x

to the left side

- 2 x = - 3 x + 2 πn + \frac{π}{2} - 70

Add

3 x

to both sides

- 2 x + 3 x = - 3 x + 2 πn + \frac{π}{2} - 70 + 3 x

Simplify

x = 2 πn + \frac{π}{2} - 70

x = 2 πn + \frac{π}{2} - 70

- (2 x - 20) = π - (\frac{π}{2} - (3 x + 50)) + 2 πn : x = - \frac{4 πn + 60 + π}{10}

- (2 x - 20) = π - (\frac{π}{2} - (3 x + 50)) + 2 πn

Expand

- (2 x - 20) : - 2 x + 20

- (2 x - 20)

Distribute parentheses

= - (2 x) - (- 20)

Apply minus-plus rules

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

= - 2 x + 20

Expand

π - (\frac{π}{2} - (3 x + 50)) + 2 πn : π - \frac{π}{2} + 3 x + 50 + 2 πn

π - (\frac{π}{2} - (3 x + 50)) + 2 πn

- (3 x + 50) : - 3 x - 50

- (3 x + 50)

Distribute parentheses

= - (3 x) - (50)

Apply minus-plus rules

+ (- a) = - a

= - 3 x - 50

= π - (- 3 x + \frac{π}{2} - 50) + 2 πn

- (\frac{π}{2} - 3 x - 50) : - \frac{π}{2} + 3 x + 50

- (\frac{π}{2} - 3 x - 50)

Distribute parentheses

= - (\frac{π}{2}) - (- 3 x) - (- 50)

Apply minus-plus rules

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

= - \frac{π}{2} + 3 x + 50

= π - \frac{π}{2} + 3 x + 50 + 2 πn

- 2 x + 20 = π - \frac{π}{2} + 3 x + 50 + 2 πn

Move

20

to the right side

- 2 x + 20 = π - \frac{π}{2} + 3 x + 50 + 2 πn

Subtract

20

from both sides

- 2 x + 20 - 20 = π - \frac{π}{2} + 3 x + 50 + 2 πn - 20

Simplify

- 2 x + 20 - 20 = π - \frac{π}{2} + 3 x + 50 + 2 πn - 20

Simplify

- 2 x + 20 - 20 : - 2 x

- 2 x + 20 - 20

Add similar elements:

20 - 20 = 0

= - 2 x

Simplify

π - \frac{π}{2} + 3 x + 50 + 2 πn - 20 : 3 x + 2 πn + 30 + π - \frac{π}{2}

π - \frac{π}{2} + 3 x + 50 + 2 πn - 20

Group like terms

= 3 x + π + 2 πn - \frac{π}{2} + 50 - 20

Add/Subtract the numbers:

50 - 20 = 30

= 3 x + 2 πn + 30 + π - \frac{π}{2}

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

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

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

Move

3 x

to the left side

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

Subtract

3 x

from both sides

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

Simplify

- 5 x = 2 πn + 30 + π - \frac{π}{2}

- 5 x = 2 πn + 30 + π - \frac{π}{2}

Divide both sides by

- 5

- 5 x = 2 πn + 30 + π - \frac{π}{2}

Divide both sides by

- 5

\frac{- 5 x}{- 5} = \frac{2 πn}{- 5} + \frac{30}{- 5} + \frac{π}{- 5} - \frac{\frac{π}{2}}{- 5}

Simplify

\frac{- 5 x}{- 5} = \frac{2 πn}{- 5} + \frac{30}{- 5} + \frac{π}{- 5} - \frac{\frac{π}{2}}{- 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{2 πn}{- 5} + \frac{30}{- 5} + \frac{π}{- 5} - \frac{\frac{π}{2}}{- 5} : - \frac{4 πn + 60 + π}{10}

\frac{2 πn}{- 5} + \frac{30}{- 5} + \frac{π}{- 5} - \frac{\frac{π}{2}}{- 5}

Apply rule

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

= \frac{2 πn + 30 + π - \frac{π}{2}}{- 5}

Apply the fraction rule:

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

= - \frac{2 πn + 30 + π - \frac{π}{2}}{5}

Join

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

2 πn + 30 + π - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + 30 \cdot 2 + π 2 - π

Add similar elements:

2 π - π = π

= 2 \cdot 2 πn + 30 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 30 \cdot 2 + π

Multiply the numbers:

30 \cdot 2 = 60

= 4 πn + 60 + π

= \frac{4 πn + 60 + π}{2}

= - \frac{\frac{4 πn + π + 60}{2}}{5}

Simplify

\frac{\frac{4 πn + 60 + π}{2}}{5} : \frac{4 πn + 60 + π}{10}

\frac{\frac{4 πn + 60 + π}{2}}{5}

Apply the fraction rule:

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

= \frac{4 πn + 60 + π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{4 πn + 60 + π}{10}

= - \frac{4 πn + π + 60}{10}

= - \frac{4 πn + 60 + π}{10}

x = - \frac{4 πn + 60 + π}{10}

x = - \frac{4 πn + 60 + π}{10}

x = - \frac{4 πn + 60 + π}{10}

x = 2 πn + \frac{π}{2} - 70, x = - \frac{4 πn + 60 + π}{10}

x = 2 πn + \frac{π}{2} - 70, x = - \frac{4 πn + 60 + π}{10}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x-20)=-cos(3x+50) ?
The general solution for sin(2x-20)=-cos(3x+50) is x=2pin+pi/2-70,x=-(4pin+60+pi)/(10)