What is the general solution for solvefor x,sin(x+60)=cos(y-37) ?

The general solution for solvefor x,sin(x+60)=cos(y-37) is x=-y+360n+67,x=y+180+360n-187

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Popular Trigonometry >

solvefor x,sin(x+60)=cos(y-37)

Solution

solve for

x, sin (x + 6 0^{\circ}) = cos (y - 3 7^{\circ})

Solution

x = - y + 36 0^{\circ} n + 6 7^{\circ}, x = y + 18 0^{\circ} + 36 0^{\circ} n - 18 7^{\circ}

Radians

x = - y + \frac{67 π}{180} + 2 πn, x = y + π - \frac{187 π}{180} + 2 πn

Solution steps

sin (x + 6 0^{\circ}) = cos (y - 3 7^{\circ})

Rewrite using trig identities

cos (y - \frac{37 π}{180})

Use the following identity:

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

sin (\frac{π}{2} - (y - \frac{37 π}{180}))

sin (x + \frac{π}{3}) = sin (\frac{π}{2} - (y - \frac{37 π}{180}))

Apply trig inverse properties

sin (x + \frac{π}{3}) = sin (\frac{π}{2} - (y - \frac{37 π}{180}))

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

x + \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn, x + \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn

x + \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn, x + \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn

x + 6 0^{\circ} = 9 0^{\circ} - (y - 3 7^{\circ}) + 36 0^{\circ} n : x = - y + 36 0^{\circ} n + 6 7^{\circ}

x + \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn

Move

6 0^{\circ}

to the right side

x + \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn

Subtract

6 0^{\circ}

from both sides

x + \frac{π}{3} - \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} = \frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn - \frac{π}{3}

Simplify

x + 6 0^{\circ} - 6 0^{\circ} : x

x + 6 0^{\circ} - 6 0^{\circ}

Add similar elements:

6 0^{\circ} - 6 0^{\circ} = 0

= x

Simplify

9 0^{\circ} - (y - 3 7^{\circ}) + 36 0^{\circ} n - 6 0^{\circ} : - y + 36 0^{\circ} n + 6 7^{\circ}

\frac{π}{2} - (y - \frac{37 π}{180}) + 2 πn - \frac{π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

9 0^{\circ} :

multiply the denominator and numerator by

3

9 0^{\circ} = \frac{18 0 ^{\circ} 3}{2 \cdot 3} = 9 0^{\circ}

For

6 0^{\circ} :

multiply the denominator and numerator by

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 6 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} 3 - 18 0 ^{\circ} 2}{6}

Add similar elements:

54 0^{\circ} - 36 0^{\circ} = 18 0^{\circ}

= - (y - \frac{37 π}{180}) + 2 πn + \frac{π}{6}

- (y - 3 7^{\circ}) : - y + 3 7^{\circ}

- (y - \frac{37 π}{180})

Distribute parentheses

= - y - (- \frac{37 π}{180})

Apply minus-plus rules

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

= - y + \frac{37 π}{180}

= - y + \frac{37 π}{180} + 2 πn + \frac{π}{6}

Simplify

- y + 3 7^{\circ} + 36 0^{\circ} n + 3 0^{\circ} : - y + 36 0^{\circ} n + 6 7^{\circ}

- y + \frac{37 π}{180} + 2 πn + \frac{π}{6}

Group like terms

= - y + 2 πn + \frac{π}{6} + \frac{37 π}{180}

Least Common Multiplier of

6, 180 : 180

6, 180

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

6

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

3 0^{\circ} :

multiply the denominator and numerator by

30

3 0^{\circ} = \frac{18 0 ^{\circ} 30}{6 \cdot 30} = 3 0^{\circ}

= 3 0^{\circ} + 3 7^{\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} 30 + 666 0 ^{\circ}}{180}

Add similar elements:

540 0^{\circ} + 666 0^{\circ} = 1206 0^{\circ}

= - y + 2 πn + \frac{67 π}{180}

= - y + 2 πn + \frac{67 π}{180}

x = - y + 2 πn + \frac{67 π}{180}

x = - y + 2 πn + \frac{67 π}{180}

x = - y + 2 πn + \frac{67 π}{180}

x + 6 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (y - 3 7^{\circ})) + 36 0^{\circ} n : x = y + 18 0^{\circ} + 36 0^{\circ} n - 18 7^{\circ}

x + \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn

Move

6 0^{\circ}

to the right side

x + \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn

Subtract

6 0^{\circ}

from both sides

x + \frac{π}{3} - \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} = π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn - \frac{π}{3}

Simplify

x + 6 0^{\circ} - 6 0^{\circ} : x

x + 6 0^{\circ} - 6 0^{\circ}

Add similar elements:

6 0^{\circ} - 6 0^{\circ} = 0

= x

Simplify

18 0^{\circ} - (9 0^{\circ} - (y - 3 7^{\circ})) + 36 0^{\circ} n - 6 0^{\circ} : y + 18 0^{\circ} + 36 0^{\circ} n - 18 7^{\circ}

π - (\frac{π}{2} - (y - \frac{37 π}{180})) + 2 πn - \frac{π}{3}

Expand

9 0^{\circ} - (y - 3 7^{\circ}) : - y + 12 7^{\circ}

\frac{π}{2} - (y - \frac{37 π}{180})

- (y - 3 7^{\circ}) : - y + 3 7^{\circ}

- (y - \frac{37 π}{180})

Distribute parentheses

= - y - (- \frac{37 π}{180})

Apply minus-plus rules

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

= - y + \frac{37 π}{180}

= \frac{π}{2} - y + \frac{37 π}{180}

Simplify

9 0^{\circ} - y + 3 7^{\circ} : - y + 12 7^{\circ}

\frac{π}{2} - y + \frac{37 π}{180}

Group like terms

= - y + \frac{π}{2} + \frac{37 π}{180}

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 7^{\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 + 666 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} + 666 0^{\circ} = 2286 0^{\circ}

= - y + \frac{127 π}{180}

= - y + \frac{127 π}{180}

= π - (- y + \frac{127 π}{180}) + 2 πn - \frac{π}{3}

- (- y + 12 7^{\circ}) : y - 12 7^{\circ}

- (- y + \frac{127 π}{180})

Distribute parentheses

= - (- y) - \frac{127 π}{180}

Apply minus-plus rules

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

= y - \frac{127 π}{180}

= π + y - \frac{127 π}{180} + 2 πn - \frac{π}{3}

Simplify

18 0^{\circ} + y - 12 7^{\circ} + 36 0^{\circ} n - 6 0^{\circ} : y + 18 0^{\circ} + 36 0^{\circ} n - 18 7^{\circ}

π + y - \frac{127 π}{180} + 2 πn - \frac{π}{3}

Group like terms

= y + π + 2 πn - \frac{π}{3} - \frac{127 π}{180}

Least Common Multiplier of

3, 180 : 180

3, 180

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

180

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

Multiply the numbers:

3 \cdot 3 \cdot 2 \cdot 2 \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

6 0^{\circ} :

multiply the denominator and numerator by

60

6 0^{\circ} = \frac{18 0 ^{\circ} 60}{3 \cdot 60} = 6 0^{\circ}

= - 6 0^{\circ} - 12 7^{\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} 60 - 2286 0 ^{\circ}}{180}

Add similar elements:

- 1080 0^{\circ} - 2286 0^{\circ} = - 3366 0^{\circ}

= \frac{- 3366 0 ^{\circ}}{180}

Apply the fraction rule:

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

= y + π + 2 πn - \frac{187 π}{180}

= y + π + 2 πn - \frac{187 π}{180}

x = y + π + 2 πn - \frac{187 π}{180}

x = y + π + 2 πn - \frac{187 π}{180}

x = y + π + 2 πn - \frac{187 π}{180}

x = - y + 36 0^{\circ} n + 6 7^{\circ}, x = y + 18 0^{\circ} + 36 0^{\circ} n - 18 7^{\circ}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,sin(x+60)=cos(y-37) ?
The general solution for solvefor x,sin(x+60)=cos(y-37) is x=-y+360n+67,x=y+180+360n-187