What is the general solution for sin(y+10)=cos(y+20) ?

The general solution for sin(y+10)=cos(y+20) is y=-360n+30,y=-150-360n

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sin(y+10)=cos(y+20)

Solution

sin (y + 1 0^{\circ}) = cos (y + 2 0^{\circ})

Solution

y = - 36 0^{\circ} n + 3 0^{\circ}, y = - 15 0^{\circ} - 36 0^{\circ} n

Radians

y = \frac{π}{6} - 2 πn, y = - \frac{5 π}{6} - 2 πn

Solution steps

sin (y + 1 0^{\circ}) = cos (y + 2 0^{\circ})

Subtract

cos (y + 2 0^{\circ})

from both sides

sin (y + 1 0^{\circ}) - cos (y + 2 0^{\circ}) = 0

Simplify

sin (y + 1 0^{\circ}) - cos (y + 2 0^{\circ}) : sin (\frac{18 y + 18 0 ^{\circ}}{18}) - cos (\frac{9 y + 18 0 ^{\circ}}{9})

sin (y + 1 0^{\circ}) - cos (y + 2 0^{\circ})

Join

y + 1 0^{\circ} : \frac{18 y + 18 0 ^{\circ}}{18}

y + 1 0^{\circ}

Convert element to fraction:

y = \frac{y 18}{18}

= \frac{y \cdot 18}{18} + 1 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{y \cdot 18 + 18 0 ^{\circ}}{18}

= sin (\frac{18 y + 18 0 ^{\circ}}{18}) - cos (y + 2 0^{\circ})

Join

y + 2 0^{\circ} : \frac{9 y + 18 0 ^{\circ}}{9}

y + 2 0^{\circ}

Convert element to fraction:

y = \frac{y 9}{9}

= \frac{y \cdot 9}{9} + 2 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{y \cdot 9 + 18 0 ^{\circ}}{9}

= sin (\frac{18 y + 18 0 ^{\circ}}{18}) - cos (\frac{9 y + 18 0 ^{\circ}}{9})

sin (\frac{18 y + 18 0 ^{\circ}}{18}) - cos (\frac{9 y + 18 0 ^{\circ}}{9}) = 0

Rewrite using trig identities

- cos (\frac{18 0 ^{\circ} + 9 y}{9}) + sin (\frac{18 0 ^{\circ} + 18 y}{18})

Use the following identity:

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

= - cos (\frac{18 0 ^{\circ} + 9 y}{9}) + cos (9 0^{\circ} - \frac{18 0 ^{\circ} + 18 y}{18})

Join

9 0^{\circ} - \frac{18 0 ^{\circ} + 18 y}{18} : \frac{72 0 ^{\circ} - 9 y}{9}

9 0^{\circ} - \frac{18 0 ^{\circ} + 18 y}{18}

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

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

2

18

= 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}

= 9 0^{\circ} - \frac{18 0 ^{\circ} + 18 y}{18}

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 - ( 18 0 ^{\circ} + 18 y )}{18}

Expand

18 0^{\circ} 9 - (18 0^{\circ} + 18 y) : 144 0^{\circ} - 18 y

18 0^{\circ} 9 - (18 0^{\circ} + 18 y)

= 162 0^{\circ} - (18 0^{\circ} + 18 y)

- (18 0^{\circ} + 18 y) : - 18 0^{\circ} - 18 y

- (18 0^{\circ} + 18 y)

Distribute parentheses

= - (18 0^{\circ}) - (18 y)

Apply minus-plus rules

+ (- a) = - a

= - 18 0^{\circ} - 18 y

= 18 0^{\circ} 9 - 18 0^{\circ} - 18 y

Add similar elements:

162 0^{\circ} - 18 0^{\circ} = 144 0^{\circ}

= 144 0^{\circ} - 18 y

= \frac{144 0 ^{\circ} - 18 y}{18}

Factor

144 0^{\circ} - 18 y : 2 (72 0^{\circ} - 9 y)

144 0^{\circ} - 18 y

Rewrite as

= 2 \cdot 72 0^{\circ} - 2 \cdot 9 y

Factor out common term

2

= 2 (72 0^{\circ} - 9 y)

= \frac{2 ( 72 0 ^{\circ} - 9 y )}{18}

Cancel the common factor:

2

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

= - cos (\frac{18 0 ^{\circ} + 9 y}{9}) + cos (\frac{72 0 ^{\circ} - 9 y}{9})

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} + \frac{18 0 ^{\circ} + 9 y}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} - \frac{18 0 ^{\circ} + 9 y}{9}}{2})

Simplify

- 2 sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} + \frac{18 0 ^{\circ} + 9 y}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} - \frac{18 0 ^{\circ} + 9 y}{9}}{2}) : - 2 sin (5 0^{\circ}) sin (\frac{- 6 y + 18 0 ^{\circ}}{6})

- 2 sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} + \frac{18 0 ^{\circ} + 9 y}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 y}{9} - \frac{18 0 ^{\circ} + 9 y}{9}}{2})

\frac{\frac{72 0 ^{\circ} - 9 y}{9} + \frac{18 0 ^{\circ} + 9 y}{9}}{2} = 5 0^{\circ}

\frac{\frac{72 0 ^{\circ} - 9 y}{9} + \frac{18 0 ^{\circ} + 9 y}{9}}{2}

Combine the fractions

\frac{- 9 y + 72 0 ^{\circ}}{9} + \frac{9 y + 18 0 ^{\circ}}{9} : 10 0^{\circ}

Apply rule

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

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

72 0^{\circ} - 9 y + 18 0^{\circ} + 9 y = 90 0^{\circ}

72 0^{\circ} - 9 y + 18 0^{\circ} + 9 y

Group like terms

= - 9 y + 9 y + 72 0^{\circ} + 18 0^{\circ}

Add similar elements:

- 9 y + 9 y = 0

= 72 0^{\circ} + 18 0^{\circ}

Add similar elements:

72 0^{\circ} + 18 0^{\circ} = 90 0^{\circ}

= 90 0^{\circ}

= 90 0^{\circ}

= \frac{10 0 ^{\circ}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

9 \cdot 2 = 18

= 5 0^{\circ}

= - 2 sin (5 0^{\circ}) sin (\frac{\frac{- 9 y + 72 0 ^{\circ}}{9} - \frac{9 y + 18 0 ^{\circ}}{9}}{2})

\frac{\frac{72 0 ^{\circ} - 9 y}{9} - \frac{18 0 ^{\circ} + 9 y}{9}}{2} = \frac{- 6 y + 18 0 ^{\circ}}{6}

\frac{\frac{72 0 ^{\circ} - 9 y}{9} - \frac{18 0 ^{\circ} + 9 y}{9}}{2}

Combine the fractions

\frac{- 9 y + 72 0 ^{\circ}}{9} - \frac{9 y + 18 0 ^{\circ}}{9} : \frac{72 0 ^{\circ} - 9 y - ( 18 0 ^{\circ} + 9 y )}{9}

Apply rule

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

= \frac{72 0 ^{\circ} - 9 y - ( 9 y + 18 0 ^{\circ} )}{9}

= \frac{\frac{72 0 ^{\circ} - 9 y - ( 9 y + 18 0 ^{\circ} )}{9}}{2}

Apply the fraction rule:

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

= \frac{72 0 ^{\circ} - 9 y - ( 18 0 ^{\circ} + 9 y )}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{72 0 ^{\circ} - 9 y - ( 9 y + 18 0 ^{\circ} )}{18}

Expand

72 0^{\circ} - 9 y - (18 0^{\circ} + 9 y) : - 18 y + 54 0^{\circ}

72 0^{\circ} - 9 y - (18 0^{\circ} + 9 y)

- (18 0^{\circ} + 9 y) : - 18 0^{\circ} - 9 y

- (18 0^{\circ} + 9 y)

Distribute parentheses

= - (18 0^{\circ}) - (9 y)

Apply minus-plus rules

+ (- a) = - a

= - 18 0^{\circ} - 9 y

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

Simplify

72 0^{\circ} - 9 y - 18 0^{\circ} - 9 y : - 18 y + 54 0^{\circ}

72 0^{\circ} - 9 y - 18 0^{\circ} - 9 y

Group like terms

= - 9 y - 9 y + 72 0^{\circ} - 18 0^{\circ}

Add similar elements:

- 9 y - 9 y = - 18 y

= - 18 y + 72 0^{\circ} - 18 0^{\circ}

Add similar elements:

72 0^{\circ} - 18 0^{\circ} = 54 0^{\circ}

= - 18 y + 54 0^{\circ}

= - 18 y + 54 0^{\circ}

= \frac{- 18 y + 54 0 ^{\circ}}{18}

Factor

- 18 y + 54 0^{\circ} : 3 (- 6 y + 18 0^{\circ})

- 18 y + 54 0^{\circ}

Rewrite as

= - 3 \cdot 6 y + 54 0^{\circ}

Factor out common term

3

= 3 (- 6 y + 18 0^{\circ})

= \frac{3 ( - 6 y + 18 0 ^{\circ} )}{18}

Cancel the common factor:

3

= \frac{- 6 y + 18 0 ^{\circ}}{6}

= - 2 sin (5 0^{\circ}) sin (\frac{- 6 y + 18 0 ^{\circ}}{6})

= - 2 sin (5 0^{\circ}) sin (\frac{- 6 y + 18 0 ^{\circ}}{6})

- 2 sin (5 0^{\circ}) sin (\frac{- 6 y + 18 0 ^{\circ}}{6}) = 0

Divide both sides by

- 2 sin (5 0^{\circ})

- 2 sin (5 0^{\circ}) sin (\frac{- 6 y + 18 0 ^{\circ}}{6}) = 0

Divide both sides by

- 2 sin (5 0^{\circ})

\frac{- 2 sin ( 5 0 ^{\circ} ) sin ( \frac{- 6 y + 18 0 ^{\circ}}{6} )}{- 2 sin ( 5 0 ^{\circ} )} = \frac{0}{- 2 sin ( 5 0 ^{\circ} )}

Simplify

sin (\frac{- 6 y + 18 0 ^{\circ}}{6}) = 0

sin (\frac{- 6 y + 18 0 ^{\circ}}{6}) = 0

General solutions for

sin (\frac{- 6 y + 18 0 ^{\circ}}{6}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

\frac{- 6 y + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n : y = - 36 0^{\circ} n + 3 0^{\circ}

\frac{- 6 y + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n

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

\frac{- 6 y + 18 0 ^{\circ}}{6} = 36 0^{\circ} n

Multiply both sides by

6

\frac{- 6 y + 18 0 ^{\circ}}{6} = 36 0^{\circ} n

Multiply both sides by

6

\frac{6 ( - 6 y + 18 0 ^{\circ} )}{6} = 6 \cdot 36 0^{\circ} n

Simplify

- 6 y + 18 0^{\circ} = 216 0^{\circ} n

- 6 y + 18 0^{\circ} = 216 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 6 y + 18 0^{\circ} = 216 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

- 6 y + 18 0^{\circ} - 18 0^{\circ} = 216 0^{\circ} n - 18 0^{\circ}

Simplify

- 6 y = 216 0^{\circ} n - 18 0^{\circ}

- 6 y = 216 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 6

- 6 y = 216 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 6

\frac{- 6 y}{- 6} = \frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

Simplify

\frac{- 6 y}{- 6} = \frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

Simplify

\frac{- 6 y}{- 6} : y

\frac{- 6 y}{- 6}

Apply the fraction rule:

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

= \frac{6 y}{6}

Divide the numbers:

\frac{6}{6} = 1

= y

Simplify

\frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6} : - 36 0^{\circ} n + 3 0^{\circ}

\frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

\frac{216 0 ^{\circ} n}{- 6} = - 36 0^{\circ} n

\frac{216 0 ^{\circ} n}{- 6}

Apply the fraction rule:

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

= - \frac{216 0 ^{\circ} n}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{18 0 ^{\circ}}{- 6}

Apply the fraction rule:

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

= - 36 0^{\circ} n - (- 3 0^{\circ})

Apply rule

- (- a) = a

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

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

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

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

Solve

\frac{- 6 y + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n : y = - 15 0^{\circ} - 36 0^{\circ} n

\frac{- 6 y + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

6

\frac{- 6 y + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

6

\frac{6 ( - 6 y + 18 0 ^{\circ} )}{6} = 108 0^{\circ} + 6 \cdot 36 0^{\circ} n

Simplify

- 6 y + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

- 6 y + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 6 y + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

- 6 y + 18 0^{\circ} - 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n - 18 0^{\circ}

Simplify

- 6 y = 90 0^{\circ} + 216 0^{\circ} n

- 6 y = 90 0^{\circ} + 216 0^{\circ} n

Divide both sides by

- 6

- 6 y = 90 0^{\circ} + 216 0^{\circ} n

Divide both sides by

- 6

\frac{- 6 y}{- 6} = \frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

Simplify

\frac{- 6 y}{- 6} = \frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

Simplify

\frac{- 6 y}{- 6} : y

\frac{- 6 y}{- 6}

Apply the fraction rule:

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

= \frac{6 y}{6}

Divide the numbers:

\frac{6}{6} = 1

= y

Simplify

\frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6} : - 15 0^{\circ} - 36 0^{\circ} n

\frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

Apply the fraction rule:

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

= - 15 0^{\circ} + \frac{216 0 ^{\circ} n}{- 6}

\frac{216 0 ^{\circ} n}{- 6} = - 36 0^{\circ} n

\frac{216 0 ^{\circ} n}{- 6}

Apply the fraction rule:

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

= - \frac{216 0 ^{\circ} n}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 36 0^{\circ} n

= - 15 0^{\circ} - 36 0^{\circ} n

y = - 15 0^{\circ} - 36 0^{\circ} n

y = - 15 0^{\circ} - 36 0^{\circ} n

y = - 15 0^{\circ} - 36 0^{\circ} n

y = - 36 0^{\circ} n + 3 0^{\circ}, y = - 15 0^{\circ} - 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(y+10)=cos(y+20) ?
The general solution for sin(y+10)=cos(y+20) is y=-360n+30,y=-150-360n