What is the general solution for sin(40-x)=cos(3x) ?

The general solution for sin(40-x)=cos(3x) is x=(6480n+900)/(36),x=-(900+6480n)/(72)

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Popular Trigonometry >

sin(40-x)=cos(3x)

Solution

sin (4 0^{\circ} - x) = cos (3 x)

Solution

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}, x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

Radians

x = \frac{5 π}{36} + \frac{36 π}{36} n, x = - \frac{5 π}{72} - \frac{36 π}{72} n

Solution steps

sin (4 0^{\circ} - x) = cos (3 x)

Rewrite using trig identities

sin (4 0^{\circ} - x) = cos (3 x)

Use the following identity:

cos (x) = sin (9 0^{\circ} - x)

sin (4 0^{\circ} - x) = sin (9 0^{\circ} - 3 x)

sin (4 0^{\circ} - x) = sin (9 0^{\circ} - 3 x)

Apply trig inverse properties

sin (4 0^{\circ} - x) = sin (9 0^{\circ} - 3 x)

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

4 0^{\circ} - x = 9 0^{\circ} - 3 x + 36 0^{\circ} n, 4 0^{\circ} - x = 18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n

4 0^{\circ} - x = 9 0^{\circ} - 3 x + 36 0^{\circ} n, 4 0^{\circ} - x = 18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n

4 0^{\circ} - x = 9 0^{\circ} - 3 x + 36 0^{\circ} n : x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

4 0^{\circ} - x = 9 0^{\circ} - 3 x + 36 0^{\circ} n

Move

4 0^{\circ}

to the right side

4 0^{\circ} - x = 9 0^{\circ} - 3 x + 36 0^{\circ} n

Subtract

4 0^{\circ}

from both sides

4 0^{\circ} - x - 4 0^{\circ} = 9 0^{\circ} - 3 x + 36 0^{\circ} n - 4 0^{\circ}

Simplify

4 0^{\circ} - x - 4 0^{\circ} = 9 0^{\circ} - 3 x + 36 0^{\circ} n - 4 0^{\circ}

Simplify

4 0^{\circ} - x - 4 0^{\circ} : - x

4 0^{\circ} - x - 4 0^{\circ}

Add similar elements:

4 0^{\circ} - 4 0^{\circ} = 0

= - x

Simplify

9 0^{\circ} - 3 x + 36 0^{\circ} n - 4 0^{\circ} : - 3 x + 36 0^{\circ} n + 5 0^{\circ}

9 0^{\circ} - 3 x + 36 0^{\circ} n - 4 0^{\circ}

Group like terms

= - 3 x + 36 0^{\circ} n + 9 0^{\circ} - 4 0^{\circ}

Least Common Multiplier of

2, 9 : 18

2, 9

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

9

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

18

For

9 0^{\circ} :

multiply the denominator and numerator by

9

9 0^{\circ} = \frac{18 0 ^{\circ} 9}{2 \cdot 9} = 9 0^{\circ}

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 9 - 72 0 ^{\circ}}{18}

Add similar elements:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= - 3 x + 36 0^{\circ} n + 5 0^{\circ}

- x = - 3 x + 36 0^{\circ} n + 5 0^{\circ}

- x = - 3 x + 36 0^{\circ} n + 5 0^{\circ}

- x = - 3 x + 36 0^{\circ} n + 5 0^{\circ}

Move

3 x

to the left side

- x = - 3 x + 36 0^{\circ} n + 5 0^{\circ}

Add

3 x

to both sides

- x + 3 x = - 3 x + 36 0^{\circ} n + 5 0^{\circ} + 3 x

Simplify

2 x = 36 0^{\circ} n + 5 0^{\circ}

2 x = 36 0^{\circ} n + 5 0^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} n + 5 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{5 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{5 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{5 0 ^{\circ}}{2} : \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

\frac{36 0 ^{\circ} n}{2} + \frac{5 0 ^{\circ}}{2}

Apply rule

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

= \frac{36 0 ^{\circ} n + 5 0 ^{\circ}}{2}

Join

36 0^{\circ} n + 5 0^{\circ} : \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{18}

36 0^{\circ} n + 5 0^{\circ}

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 18}{18}

= \frac{36 0 ^{\circ} n \cdot 18}{18} + 5 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{36 0 ^{\circ} n \cdot 18 + 90 0 ^{\circ}}{18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{18}

= \frac{\frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{18}}{2}

Apply the fraction rule:

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

= \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}

4 0^{\circ} - x = 18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n : x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

4 0^{\circ} - x = 18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n

Expand

18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n : 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n

- (9 0^{\circ} - 3 x) : - 9 0^{\circ} + 3 x

- (9 0^{\circ} - 3 x)

Distribute parentheses

= - (9 0^{\circ}) - (- 3 x)

Apply minus-plus rules

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

= - 9 0^{\circ} + 3 x

= 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n

4 0^{\circ} - x = 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n

Move

4 0^{\circ}

to the right side

4 0^{\circ} - x = 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n

Subtract

4 0^{\circ}

from both sides

4 0^{\circ} - x - 4 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n - 4 0^{\circ}

Simplify

4 0^{\circ} - x - 4 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n - 4 0^{\circ}

Simplify

4 0^{\circ} - x - 4 0^{\circ} : - x

4 0^{\circ} - x - 4 0^{\circ}

Add similar elements:

4 0^{\circ} - 4 0^{\circ} = 0

= - x

Simplify

18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n - 4 0^{\circ} : 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

18 0^{\circ} - 9 0^{\circ} + 3 x + 36 0^{\circ} n - 4 0^{\circ}

Group like terms

= 3 x + 18 0^{\circ} + 36 0^{\circ} n - 9 0^{\circ} - 4 0^{\circ}

Least Common Multiplier of

2, 9 : 18

2, 9

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

9

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

18

For

9 0^{\circ} :

multiply the denominator and numerator by

9

9 0^{\circ} = \frac{18 0 ^{\circ} 9}{2 \cdot 9} = 9 0^{\circ}

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= - 9 0^{\circ} - 4 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{- 18 0 ^{\circ} 9 - 72 0 ^{\circ}}{18}

Add similar elements:

- 162 0^{\circ} - 72 0^{\circ} = - 234 0^{\circ}

= \frac{- 234 0 ^{\circ}}{18}

Apply the fraction rule:

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

= 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

- x = 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

- x = 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

- x = 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

Move

3 x

to the left side

- x = 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

Subtract

3 x

from both sides

- x - 3 x = 3 x + 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ} - 3 x

Simplify

- 4 x = 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

- 4 x = 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

Divide both sides by

- 4

- 4 x = 18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

Divide both sides by

- 4

\frac{- 4 x}{- 4} = \frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{13 0 ^{\circ}}{- 4}

Simplify

\frac{- 4 x}{- 4} = \frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{13 0 ^{\circ}}{- 4}

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{13 0 ^{\circ}}{- 4} : - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

\frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{13 0 ^{\circ}}{- 4}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 13 0 ^{\circ}}{- 4}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 13 0 ^{\circ}}{4}

Join

18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ} : \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}

18 0^{\circ} + 36 0^{\circ} n - 13 0^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 18}{18}

= 18 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 18}{18} - 13 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 18 + 36 0 ^{\circ} n \cdot 18 - 234 0 ^{\circ}}{18}

18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18 - 234 0^{\circ} = 90 0^{\circ} + 648 0^{\circ} n

18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18 - 234 0^{\circ}

Add similar elements:

324 0^{\circ} - 234 0^{\circ} = 90 0^{\circ}

= 90 0^{\circ} + 2 \cdot 324 0^{\circ} n

Multiply the numbers:

2 \cdot 18 = 36

= 90 0^{\circ} + 648 0^{\circ} n

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}

= - \frac{\frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}}{4}

Simplify

\frac{\frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}}{4} : \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

\frac{\frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}}{4}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18 \cdot 4}

Multiply the numbers:

18 \cdot 4 = 72

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

= - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}, x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

x = \frac{648 0 ^{\circ} n + 90 0 ^{\circ}}{36}, x = - \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{72}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(40-x)=cos(3x) ?
The general solution for sin(40-x)=cos(3x) is x=(6480n+900)/(36),x=-(900+6480n)/(72)