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Popular Trigonometry >

sin(5x+43)=cos(-x+31)

Solution

sin (5 x + 43) = cos (- x + 3 1^{\circ})

Solution

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}, x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

Radians

x = \frac{\frac{59 π}{5}}{144} - \frac{43}{4} + \frac{360 π}{720} n, x = - \frac{43}{6} + \frac{\frac{121 π}{5}}{216} + \frac{360 π}{1080} n

Solution steps

sin (5 x + 43) = cos (- x + 3 1^{\circ})

Rewrite using trig identities

sin (5 x + 43) = cos (- x + 3 1^{\circ})

Use the following identity:

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

sin (5 x + 43) = sin (9 0^{\circ} - (- x + 3 1^{\circ}))

sin (5 x + 43) = sin (9 0^{\circ} - (- x + 3 1^{\circ}))

Apply trig inverse properties

sin (5 x + 43) = sin (9 0^{\circ} - (- x + 3 1^{\circ}))

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

5 x + 43 = 9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n, 5 x + 43 = 18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n

5 x + 43 = 9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n, 5 x + 43 = 18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n

5 x + 43 = 9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n : x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

5 x + 43 = 9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n

Expand

9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n : x + 36 0^{\circ} n + 5 9^{\circ}

9 0^{\circ} - (- x + 3 1^{\circ}) + 36 0^{\circ} n

- (- x + 3 1^{\circ}) : x - 3 1^{\circ}

- (- x + 3 1^{\circ})

Distribute parentheses

= - (- x) - (3 1^{\circ})

Apply minus-plus rules

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

= x - 3 1^{\circ}

= 9 0^{\circ} + x - 3 1^{\circ} + 36 0^{\circ} n

Simplify

9 0^{\circ} + x - 3 1^{\circ} + 36 0^{\circ} n : x + 36 0^{\circ} n + 5 9^{\circ}

9 0^{\circ} + x - 3 1^{\circ} + 36 0^{\circ} n

Group like terms

= x + 36 0^{\circ} n + 9 0^{\circ} - 3 1^{\circ}

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

180 : 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

180

180

divides by

2180 = 90 \cdot 2

= 2 \cdot 90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

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

2

180

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

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

180

For

9 0^{\circ} :

multiply the denominator and numerator by

90

9 0^{\circ} = \frac{18 0 ^{\circ} 90}{2 \cdot 90} = 9 0^{\circ}

= 9 0^{\circ} - 3 1^{\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} 90 - 558 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 558 0^{\circ} = 1062 0^{\circ}

= x + 36 0^{\circ} n + 5 9^{\circ}

= x + 36 0^{\circ} n + 5 9^{\circ}

5 x + 43 = x + 36 0^{\circ} n + 5 9^{\circ}

Move

43

to the right side

5 x + 43 = x + 36 0^{\circ} n + 5 9^{\circ}

Subtract

43

from both sides

5 x + 43 - 43 = x + 36 0^{\circ} n + 5 9^{\circ} - 43

Simplify

5 x = x + 36 0^{\circ} n + 5 9^{\circ} - 43

5 x = x + 36 0^{\circ} n + 5 9^{\circ} - 43

Move

x

to the left side

5 x = x + 36 0^{\circ} n + 5 9^{\circ} - 43

Subtract

x

from both sides

5 x - x = x + 36 0^{\circ} n + 5 9^{\circ} - 43 - x

Simplify

4 x = 36 0^{\circ} n + 5 9^{\circ} - 43

4 x = 36 0^{\circ} n + 5 9^{\circ} - 43

Divide both sides by

4

4 x = 36 0^{\circ} n + 5 9^{\circ} - 43

Divide both sides by

4

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{5 9 ^{\circ}}{4} - \frac{43}{4}

Simplify

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{5 9 ^{\circ}}{4} - \frac{43}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{4} + \frac{5 9 ^{\circ}}{4} - \frac{43}{4} : \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

\frac{36 0 ^{\circ} n}{4} + \frac{5 9 ^{\circ}}{4} - \frac{43}{4}

Apply rule

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

= \frac{36 0 ^{\circ} n + 5 9 ^{\circ} - 43}{4}

Join

36 0^{\circ} n + 5 9^{\circ} - 43 : \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{180}

36 0^{\circ} n + 5 9^{\circ} - 43

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 180}{180}, 43 = \frac{43 \cdot 180}{180}

= \frac{36 0 ^{\circ} n \cdot 180}{180} + 5 9^{\circ} - \frac{43 \cdot 180}{180}

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 180 + 1062 0 ^{\circ} - 43 \cdot 180}{180}

36 0^{\circ} n \cdot 180 + 1062 0^{\circ} - 43 \cdot 180 = 6480 0^{\circ} n + 1062 0^{\circ} - 7740

36 0^{\circ} n \cdot 180 + 1062 0^{\circ} - 43 \cdot 180

Multiply the numbers:

2 \cdot 180 = 360

= 6480 0^{\circ} n + 1062 0^{\circ} - 43 \cdot 180

Multiply the numbers:

43 \cdot 180 = 7740

= 6480 0^{\circ} n + 1062 0^{\circ} - 7740

= \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{180}

= \frac{\frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{180}}{4}

Apply the fraction rule:

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

= \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{180 \cdot 4}

Multiply the numbers:

180 \cdot 4 = 720

= \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}

5 x + 43 = 18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n : x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

5 x + 43 = 18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n

Expand

18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (- x + 3 1^{\circ})) + 36 0^{\circ} n

Expand

9 0^{\circ} - (- x + 3 1^{\circ}) : x + 5 9^{\circ}

9 0^{\circ} - (- x + 3 1^{\circ})

- (- x + 3 1^{\circ}) : x - 3 1^{\circ}

- (- x + 3 1^{\circ})

Distribute parentheses

= - (- x) - (3 1^{\circ})

Apply minus-plus rules

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

= x - 3 1^{\circ}

= 9 0^{\circ} + x - 3 1^{\circ}

Simplify

9 0^{\circ} + x - 3 1^{\circ} : x + 5 9^{\circ}

9 0^{\circ} + x - 3 1^{\circ}

Group like terms

= x + 9 0^{\circ} - 3 1^{\circ}

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

180 : 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

180

180

divides by

2180 = 90 \cdot 2

= 2 \cdot 90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

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

2

180

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

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

180

For

9 0^{\circ} :

multiply the denominator and numerator by

90

9 0^{\circ} = \frac{18 0 ^{\circ} 90}{2 \cdot 90} = 9 0^{\circ}

= 9 0^{\circ} - 3 1^{\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} 90 - 558 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 558 0^{\circ} = 1062 0^{\circ}

= x + 5 9^{\circ}

= x + 5 9^{\circ}

= 18 0^{\circ} - (x + 5 9^{\circ}) + 36 0^{\circ} n

- (x + 5 9^{\circ}) : - x - 5 9^{\circ}

- (x + 5 9^{\circ})

Distribute parentheses

= - (x) - (5 9^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - x - 5 9^{\circ}

= 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n

5 x + 43 = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n

Move

43

to the right side

5 x + 43 = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n

Subtract

43

from both sides

5 x + 43 - 43 = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n - 43

Simplify

5 x = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n - 43

5 x = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n - 43

Move

x

to the left side

5 x = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n - 43

Add

x

to both sides

5 x + x = 18 0^{\circ} - x - 5 9^{\circ} + 36 0^{\circ} n - 43 + x

Simplify

6 x = 18 0^{\circ} - 5 9^{\circ} + 36 0^{\circ} n - 43

6 x = 18 0^{\circ} - 5 9^{\circ} + 36 0^{\circ} n - 43

Divide both sides by

6

6 x = 18 0^{\circ} - 5 9^{\circ} + 36 0^{\circ} n - 43

Divide both sides by

6

\frac{6 x}{6} = 3 0^{\circ} - \frac{5 9 ^{\circ}}{6} + \frac{36 0 ^{\circ} n}{6} - \frac{43}{6}

Simplify

\frac{6 x}{6} = 3 0^{\circ} - \frac{5 9 ^{\circ}}{6} + \frac{36 0 ^{\circ} n}{6} - \frac{43}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

3 0^{\circ} - \frac{5 9 ^{\circ}}{6} + \frac{36 0 ^{\circ} n}{6} - \frac{43}{6} : \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

3 0^{\circ} - \frac{5 9 ^{\circ}}{6} + \frac{36 0 ^{\circ} n}{6} - \frac{43}{6}

Group like terms

= 3 0^{\circ} - \frac{43}{6} + \frac{36 0 ^{\circ} n}{6} - \frac{5 9 ^{\circ}}{6}

Apply rule

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

= \frac{18 0 ^{\circ} - 43 + 36 0 ^{\circ} n - 5 9 ^{\circ}}{6}

Join

18 0^{\circ} - 43 + 36 0^{\circ} n - 5 9^{\circ} : \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{180}

18 0^{\circ} - 43 + 36 0^{\circ} n - 5 9^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 43 = \frac{43 \cdot 180}{180}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 180}{180}

= 18 0^{\circ} - \frac{43 \cdot 180}{180} + \frac{36 0 ^{\circ} n \cdot 180}{180} - 5 9^{\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} 180 - 43 \cdot 180 + 36 0 ^{\circ} n \cdot 180 - 1062 0 ^{\circ}}{180}

18 0^{\circ} 180 - 43 \cdot 180 + 36 0^{\circ} n \cdot 180 - 1062 0^{\circ} = 2178 0^{\circ} + 6480 0^{\circ} n - 7740

18 0^{\circ} 180 - 43 \cdot 180 + 36 0^{\circ} n \cdot 180 - 1062 0^{\circ}

Group like terms

= 3240 0^{\circ} - 1062 0^{\circ} + 2 \cdot 3240 0^{\circ} n - 43 \cdot 180

Add similar elements:

3240 0^{\circ} - 1062 0^{\circ} = 2178 0^{\circ}

= 2178 0^{\circ} + 2 \cdot 3240 0^{\circ} n - 43 \cdot 180

Multiply the numbers:

2 \cdot 180 = 360

= 2178 0^{\circ} + 6480 0^{\circ} n - 43 \cdot 180

Multiply the numbers:

43 \cdot 180 = 7740

= 2178 0^{\circ} + 6480 0^{\circ} n - 7740

= \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{180}

= \frac{\frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{180}}{6}

Apply the fraction rule:

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

= \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{180 \cdot 6}

Multiply the numbers:

180 \cdot 6 = 1080

= \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}, x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

x = \frac{6480 0 ^{\circ} n + 1062 0 ^{\circ} - 7740}{720}, x = \frac{2178 0 ^{\circ} + 6480 0 ^{\circ} n - 7740}{1080}

Graph

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