What is the general solution for sin(x+20)=cos(x-50) ?

The general solution for sin(x+20)=cos(x-50) is x=-360n+60,x=-120-360n

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Popular Trigonometry >

sin(x+20)=cos(x-50)

Solution

sin (x + 2 0^{\circ}) = cos (x - 5 0^{\circ})

Solution

x = - 36 0^{\circ} n + 6 0^{\circ}, x = - 12 0^{\circ} - 36 0^{\circ} n

Radians

x = \frac{π}{3} - 2 πn, x = - \frac{2 π}{3} - 2 πn

Solution steps

sin (x + 2 0^{\circ}) = cos (x - 5 0^{\circ})

Subtract

cos (x - 5 0^{\circ})

from both sides

sin (x + 2 0^{\circ}) - cos (x - 5 0^{\circ}) = 0

Simplify

sin (x + 2 0^{\circ}) - cos (x - 5 0^{\circ}) : sin (\frac{9 x + 18 0 ^{\circ}}{9}) - cos (\frac{18 x - 90 0 ^{\circ}}{18})

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

Join

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

x + 2 0^{\circ}

Convert element to fraction:

x = \frac{x 9}{9}

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

= sin (\frac{9 x + 18 0 ^{\circ}}{9}) - cos (x - 5 0^{\circ})

Join

x - 5 0^{\circ} : \frac{18 x - 90 0 ^{\circ}}{18}

x - 5 0^{\circ}

Convert element to fraction:

x = \frac{x 18}{18}

= \frac{x \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{x \cdot 18 - 90 0 ^{\circ}}{18}

= sin (\frac{9 x + 18 0 ^{\circ}}{9}) - cos (\frac{18 x - 90 0 ^{\circ}}{18})

sin (\frac{9 x + 18 0 ^{\circ}}{9}) - cos (\frac{18 x - 90 0 ^{\circ}}{18}) = 0

Rewrite using trig identities

- cos (\frac{18 x - 90 0 ^{\circ}}{18}) + sin (\frac{18 0 ^{\circ} + 9 x}{9})

Use the following identity:

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

= - cos (\frac{18 x - 90 0 ^{\circ}}{18}) + cos (9 0^{\circ} - \frac{18 0 ^{\circ} + 9 x}{9})

Join

9 0^{\circ} - \frac{18 0 ^{\circ} + 9 x}{9} : \frac{126 0 ^{\circ} - 18 x}{18}

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

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

\frac{18 0 ^{\circ} + 9 x}{9} :

multiply the denominator and numerator by

2

\frac{18 0 ^{\circ} + 9 x}{9} = \frac{( 18 0 ^{\circ} + 9 x ) \cdot 2}{9 \cdot 2} = \frac{( 18 0 ^{\circ} + 9 x ) \cdot 2}{18}

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

Expand

18 0^{\circ} 9 - (18 0^{\circ} + 9 x) \cdot 2 : 126 0^{\circ} - 18 x

18 0^{\circ} 9 - (18 0^{\circ} + 9 x) \cdot 2

= 162 0^{\circ} - 2 (18 0^{\circ} + 9 x)

Expand

- 2 (18 0^{\circ} + 9 x) : - 36 0^{\circ} - 18 x

- 2 (18 0^{\circ} + 9 x)

Apply the distributive law:

a (b + c) = ab + a c

a = - 2, b = 18 0^{\circ}, c = 9 x

= - 36 0^{\circ} + (- 2) \cdot 9 x

Apply minus-plus rules

+ (- a) = - a

= - 36 0^{\circ} - 2 \cdot 9 x

Multiply the numbers:

2 \cdot 9 = 18

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

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

Add similar elements:

162 0^{\circ} - 36 0^{\circ} = 126 0^{\circ}

= 126 0^{\circ} - 18 x

= \frac{126 0 ^{\circ} - 18 x}{18}

= - cos (\frac{18 x - 90 0 ^{\circ}}{18}) + cos (\frac{126 0 ^{\circ} - 18 x}{18})

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{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18}}{2}) sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2})

Simplify

- 2 sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18}}{2}) sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2}) : - 2 sin (1 0^{\circ}) sin (\frac{- 3 x + 18 0 ^{\circ}}{3})

- 2 sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18}}{2}) sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2})

\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18}}{2} = 1 0^{\circ}

\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18}}{2}

Combine the fractions

\frac{- 18 x + 126 0 ^{\circ}}{18} + \frac{18 x - 90 0 ^{\circ}}{18} : 2 0^{\circ}

Apply rule

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

= \frac{- 18 x + 126 0 ^{\circ} + 18 x - 90 0 ^{\circ}}{18}

- 18 x + 126 0^{\circ} + 18 x - 90 0^{\circ} = 36 0^{\circ}

- 18 x + 126 0^{\circ} + 18 x - 90 0^{\circ}

Group like terms

= - 18 x + 18 x + 126 0^{\circ} - 90 0^{\circ}

Add similar elements:

- 18 x + 18 x = 0

= 126 0^{\circ} - 90 0^{\circ}

Add similar elements:

126 0^{\circ} - 90 0^{\circ} = 36 0^{\circ}

= 36 0^{\circ}

= 2 0^{\circ}

Cancel the common factor:

2

= 2 0^{\circ}

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

Apply the fraction rule:

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

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

Multiply the numbers:

9 \cdot 2 = 18

= 1 0^{\circ}

= - 2 sin (1 0^{\circ}) sin (\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2})

\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2} = \frac{- 3 x + 18 0 ^{\circ}}{3}

\frac{\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18}}{2}

Combine the fractions

\frac{- 18 x + 126 0 ^{\circ}}{18} - \frac{18 x - 90 0 ^{\circ}}{18} : \frac{- 18 x + 126 0 ^{\circ} - ( 18 x - 90 0 ^{\circ} )}{18}

Apply rule

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

= \frac{- 18 x + 126 0 ^{\circ} - ( 18 x - 90 0 ^{\circ} )}{18}

= \frac{\frac{- 18 x + 126 0 ^{\circ} - ( 18 x - 90 0 ^{\circ} )}{18}}{2}

Apply the fraction rule:

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

= \frac{- 18 x + 126 0 ^{\circ} - ( 18 x - 90 0 ^{\circ} )}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= \frac{- 18 x + 126 0 ^{\circ} - ( 18 x - 90 0 ^{\circ} )}{36}

Expand

- 18 x + 126 0^{\circ} - (18 x - 90 0^{\circ}) : - 36 x + 216 0^{\circ}

- 18 x + 126 0^{\circ} - (18 x - 90 0^{\circ})

- (18 x - 90 0^{\circ}) : - 18 x + 90 0^{\circ}

- (18 x - 90 0^{\circ})

Distribute parentheses

= - (18 x) - (- 90 0^{\circ})

Apply minus-plus rules

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

= - 18 x + 90 0^{\circ}

= - 18 x + 126 0^{\circ} - 18 x + 90 0^{\circ}

Simplify

- 18 x + 126 0^{\circ} - 18 x + 90 0^{\circ} : - 36 x + 216 0^{\circ}

- 18 x + 126 0^{\circ} - 18 x + 90 0^{\circ}

Group like terms

= - 18 x - 18 x + 126 0^{\circ} + 90 0^{\circ}

Add similar elements:

- 18 x - 18 x = - 36 x

= - 36 x + 126 0^{\circ} + 90 0^{\circ}

Add similar elements:

126 0^{\circ} + 90 0^{\circ} = 216 0^{\circ}

= - 36 x + 216 0^{\circ}

= - 36 x + 216 0^{\circ}

= \frac{- 36 x + 216 0 ^{\circ}}{36}

Factor

- 36 x + 216 0^{\circ} : 12 (- 3 x + 18 0^{\circ})

- 36 x + 216 0^{\circ}

Rewrite as

= - 12 \cdot 3 x + 216 0^{\circ}

Factor out common term

12

= 12 (- 3 x + 18 0^{\circ})

= \frac{12 ( - 3 x + 18 0 ^{\circ} )}{36}

Cancel the common factor:

12

= \frac{- 3 x + 18 0 ^{\circ}}{3}

= - 2 sin (1 0^{\circ}) sin (\frac{- 3 x + 18 0 ^{\circ}}{3})

= - 2 sin (1 0^{\circ}) sin (\frac{- 3 x + 18 0 ^{\circ}}{3})

- 2 sin (1 0^{\circ}) sin (\frac{- 3 x + 18 0 ^{\circ}}{3}) = 0

Divide both sides by

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

- 2 sin (1 0^{\circ}) sin (\frac{- 3 x + 18 0 ^{\circ}}{3}) = 0

Divide both sides by

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

\frac{- 2 sin ( 1 0 ^{\circ} ) sin ( \frac{- 3 x + 18 0 ^{\circ}}{3} )}{- 2 sin ( 1 0 ^{\circ} )} = \frac{0}{- 2 sin ( 1 0 ^{\circ} )}

Simplify

sin (\frac{- 3 x + 18 0 ^{\circ}}{3}) = 0

sin (\frac{- 3 x + 18 0 ^{\circ}}{3}) = 0

General solutions for

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

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

Solve

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

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

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

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

- 3 x + 18 0^{\circ} = 108 0^{\circ} n

- 3 x + 18 0^{\circ} = 108 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 3 x + 18 0^{\circ} = 108 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

- 3 x + 18 0^{\circ} - 18 0^{\circ} = 108 0^{\circ} n - 18 0^{\circ}

Simplify

- 3 x = 108 0^{\circ} n - 18 0^{\circ}

- 3 x = 108 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 3

- 3 x = 108 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 3

\frac{- 3 x}{- 3} = \frac{108 0 ^{\circ} n}{- 3} - \frac{18 0 ^{\circ}}{- 3}

Simplify

\frac{- 3 x}{- 3} = \frac{108 0 ^{\circ} n}{- 3} - \frac{18 0 ^{\circ}}{- 3}

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{108 0 ^{\circ} n}{- 3} - \frac{18 0 ^{\circ}}{- 3} : - 36 0^{\circ} n + 6 0^{\circ}

\frac{108 0 ^{\circ} n}{- 3} - \frac{18 0 ^{\circ}}{- 3}

\frac{108 0 ^{\circ} n}{- 3} = - 36 0^{\circ} n

\frac{108 0 ^{\circ} n}{- 3}

Apply the fraction rule:

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

= - \frac{108 0 ^{\circ} n}{3}

Divide the numbers:

\frac{6}{3} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{18 0 ^{\circ}}{- 3}

Apply the fraction rule:

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

= - 36 0^{\circ} n - (- 6 0^{\circ})

Apply rule

- (- a) = a

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

x = - 36 0^{\circ} n + 6 0^{\circ}

x = - 36 0^{\circ} n + 6 0^{\circ}

x = - 36 0^{\circ} n + 6 0^{\circ}

Solve

\frac{- 3 x + 18 0 ^{\circ}}{3} = 18 0^{\circ} + 36 0^{\circ} n : x = - 12 0^{\circ} - 36 0^{\circ} n

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

Multiply both sides by

3

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

Multiply both sides by

3

\frac{3 ( - 3 x + 18 0 ^{\circ} )}{3} = 54 0^{\circ} + 3 \cdot 36 0^{\circ} n

Simplify

- 3 x + 18 0^{\circ} = 54 0^{\circ} + 108 0^{\circ} n

- 3 x + 18 0^{\circ} = 54 0^{\circ} + 108 0^{\circ} n

Move

18 0^{\circ}

to the right side

- 3 x + 18 0^{\circ} = 54 0^{\circ} + 108 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

- 3 x + 18 0^{\circ} - 18 0^{\circ} = 54 0^{\circ} + 108 0^{\circ} n - 18 0^{\circ}

Simplify

- 3 x = 36 0^{\circ} + 108 0^{\circ} n

- 3 x = 36 0^{\circ} + 108 0^{\circ} n

Divide both sides by

- 3

- 3 x = 36 0^{\circ} + 108 0^{\circ} n

Divide both sides by

- 3

\frac{- 3 x}{- 3} = \frac{36 0 ^{\circ}}{- 3} + \frac{108 0 ^{\circ} n}{- 3}

Simplify

\frac{- 3 x}{- 3} = \frac{36 0 ^{\circ}}{- 3} + \frac{108 0 ^{\circ} n}{- 3}

Simplify

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

Apply the fraction rule:

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

= \frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ}}{- 3} + \frac{108 0 ^{\circ} n}{- 3} : - 12 0^{\circ} - 36 0^{\circ} n

\frac{36 0 ^{\circ}}{- 3} + \frac{108 0 ^{\circ} n}{- 3}

Apply the fraction rule:

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

= - 12 0^{\circ} + \frac{108 0 ^{\circ} n}{- 3}

\frac{108 0 ^{\circ} n}{- 3} = - 36 0^{\circ} n

\frac{108 0 ^{\circ} n}{- 3}

Apply the fraction rule:

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

= - \frac{108 0 ^{\circ} n}{3}

Divide the numbers:

\frac{6}{3} = 2

= - 36 0^{\circ} n

= - 12 0^{\circ} - 36 0^{\circ} n

x = - 12 0^{\circ} - 36 0^{\circ} n

x = - 12 0^{\circ} - 36 0^{\circ} n

x = - 12 0^{\circ} - 36 0^{\circ} n

x = - 36 0^{\circ} n + 6 0^{\circ}, x = - 12 0^{\circ} - 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+20)=cos(x-50) ?
The general solution for sin(x+20)=cos(x-50) is x=-360n+60,x=-120-360n