What is the general solution for sin(9x+40)=cos(-5x+42) ?

The general solution for sin(9x+40)=cos(-5x+42) is x=(4pin+pi-164)/8 ,x=(4pin+4+pi)/(28)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

sin(9x+40)=cos(-5x+42)

Solution

sin (9 x + 40) = cos (- 5 x + 42)

Solution

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

Degrees

x = - 1152.06348 \dots^{\circ} + 9 0^{\circ} n, x = 14.61368 \dots^{\circ} + 25.71428 \dots^{\circ} n

Solution steps

sin (9 x + 40) = cos (- 5 x + 42)

Rewrite using trig identities

sin (9 x + 40) = cos (- 5 x + 42)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

Apply trig inverse properties

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

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

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn, 9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn, 9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn : x = \frac{4 πn + π - 164}{8}

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn

Expand

\frac{π}{2} - (- 5 x + 42) + 2 πn : \frac{π}{2} + 5 x - 42 + 2 πn

\frac{π}{2} - (- 5 x + 42) + 2 πn

- (- 5 x + 42) : 5 x - 42

- (- 5 x + 42)

Distribute parentheses

= - (- 5 x) - (42)

Apply minus-plus rules

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

= 5 x - 42

= \frac{π}{2} + 5 x - 42 + 2 πn

9 x + 40 = \frac{π}{2} + 5 x - 42 + 2 πn

Move

40

to the right side

9 x + 40 = \frac{π}{2} + 5 x - 42 + 2 πn

Subtract

40

from both sides

9 x + 40 - 40 = \frac{π}{2} + 5 x - 42 + 2 πn - 40

Simplify

9 x + 40 - 40 = \frac{π}{2} + 5 x - 42 + 2 πn - 40

Simplify

9 x + 40 - 40 : 9 x

9 x + 40 - 40

Add similar elements:

40 - 40 = 0

= 9 x

Simplify

\frac{π}{2} + 5 x - 42 + 2 πn - 40 : 5 x + 2 πn + \frac{π}{2} - 82

\frac{π}{2} + 5 x - 42 + 2 πn - 40

Group like terms

= 5 x + 2 πn + \frac{π}{2} - 42 - 40

Subtract the numbers:

- 42 - 40 = - 82

= 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

Move

5 x

to the left side

9 x = 5 x + 2 πn + \frac{π}{2} - 82

Subtract

5 x

from both sides

9 x - 5 x = 5 x + 2 πn + \frac{π}{2} - 82 - 5 x

Simplify

4 x = 2 πn + \frac{π}{2} - 82

4 x = 2 πn + \frac{π}{2} - 82

Divide both sides by

4

4 x = 2 πn + \frac{π}{2} - 82

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

Simplify

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4} : \frac{4 πn + π - 164}{8}

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

Apply rule

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

= \frac{2 πn + \frac{π}{2} - 82}{4}

Join

2 πn + \frac{π}{2} - 82 : \frac{4 πn + π - 164}{2}

2 πn + \frac{π}{2} - 82

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}, 82 = \frac{82 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{82 \cdot 2}{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 - 82 \cdot 2}{2}

2 πn \cdot 2 + π - 82 \cdot 2 = 4 πn + π - 164

2 πn \cdot 2 + π - 82 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + π - 82 \cdot 2

Multiply the numbers:

82 \cdot 2 = 164

= 4 πn + π - 164

= \frac{4 πn + π - 164}{2}

= \frac{\frac{4 πn + π - 164}{2}}{4}

Apply the fraction rule:

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

= \frac{4 πn + π - 164}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn : x = \frac{4 πn + 4 + π}{28}

9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

Expand

π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn : π - \frac{π}{2} - 5 x + 42 + 2 πn

π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

- (- 5 x + 42) : 5 x - 42

- (- 5 x + 42)

Distribute parentheses

= - (- 5 x) - (42)

Apply minus-plus rules

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

= 5 x - 42

= π - (5 x + \frac{π}{2} - 42) + 2 πn

- (\frac{π}{2} + 5 x - 42) : - \frac{π}{2} - 5 x + 42

- (\frac{π}{2} + 5 x - 42)

Distribute parentheses

= - (\frac{π}{2}) - (5 x) - (- 42)

Apply minus-plus rules

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

= - \frac{π}{2} - 5 x + 42

= π - \frac{π}{2} - 5 x + 42 + 2 πn

9 x + 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn

Move

40

to the right side

9 x + 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn

Subtract

40

from both sides

9 x + 40 - 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

Simplify

9 x + 40 - 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

Simplify

9 x + 40 - 40 : 9 x

9 x + 40 - 40

Add similar elements:

40 - 40 = 0

= 9 x

Simplify

π - \frac{π}{2} - 5 x + 42 + 2 πn - 40 : - 5 x + 2 πn + 2 + π - \frac{π}{2}

π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

Group like terms

= - 5 x + π + 2 πn - \frac{π}{2} + 42 - 40

Add/Subtract the numbers:

42 - 40 = 2

= - 5 x + 2 πn + 2 + π - \frac{π}{2}

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

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

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

Move

5 x

to the left side

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

Add

5 x

to both sides

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

Simplify

14 x = 2 πn + 2 + π - \frac{π}{2}

14 x = 2 πn + 2 + π - \frac{π}{2}

Divide both sides by

14

14 x = 2 πn + 2 + π - \frac{π}{2}

Divide both sides by

14

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

Simplify

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

Simplify

\frac{14 x}{14} : x

\frac{14 x}{14}

Divide the numbers:

\frac{14}{14} = 1

= x

Simplify

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

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

Apply rule

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

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

Join

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

2 πn + 2 + π - \frac{π}{2}

Convert element to fraction:

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

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

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

2 πn \cdot 2 + 2 \cdot 2 + π 2 - π

Add similar elements:

2 π - π = π

= 2 \cdot 2 πn + 2 \cdot 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn + 4 + π

= \frac{4 πn + 4 + π}{2}

= \frac{\frac{4 πn + 4 + π}{2}}{14}

Apply the fraction rule:

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

= \frac{4 πn + 4 + π}{2 \cdot 14}

Multiply the numbers:

2 \cdot 14 = 28

= \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

Graph

Popular Examples

sin^2(a)=-((5sqrt(11)))/((18))

3.87sin((2pi(t+101.75))/(365))+11.7=14

2sin(2x)=sqrt(3),0<= x<= 2pi

cosh(2x)+sinh^2(x)-13sinh(x)=-3

sin(3x-pi/4)=1

Frequently Asked Questions (FAQ)

What is the general solution for sin(9x+40)=cos(-5x+42) ?
The general solution for sin(9x+40)=cos(-5x+42) is x=(4pin+pi-164)/8 ,x=(4pin+4+pi)/(28)