What is the general solution for sin(x-15)=cos(20+x) ?

The general solution for sin(x-15)=cos(20+x) is x=(12960n+3060)/(72)

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Popular Trigonometry >

sin(x-15)=cos(20+x)

Solution

sin (x - 1 5^{\circ}) = cos (2 0^{\circ} + x)

Solution

x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

Radians

x = \frac{17 π}{72} + \frac{72 π}{72} n

Solution steps

sin (x - 1 5^{\circ}) = cos (2 0^{\circ} + x)

Rewrite using trig identities

sin (x - 1 5^{\circ}) = cos (2 0^{\circ} + x)

Use the following identity:

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

sin (x - 1 5^{\circ}) = sin (9 0^{\circ} - (2 0^{\circ} + x))

sin (x - 1 5^{\circ}) = sin (9 0^{\circ} - (2 0^{\circ} + x))

Apply trig inverse properties

sin (x - 1 5^{\circ}) = sin (9 0^{\circ} - (2 0^{\circ} + x))

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

x - 1 5^{\circ} = 9 0^{\circ} - (2 0^{\circ} + x) + 36 0^{\circ} n, x - 1 5^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 0^{\circ} + x)) + 36 0^{\circ} n

x - 1 5^{\circ} = 9 0^{\circ} - (2 0^{\circ} + x) + 36 0^{\circ} n, x - 1 5^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 0^{\circ} + x)) + 36 0^{\circ} n

x - 1 5^{\circ} = 9 0^{\circ} - (2 0^{\circ} + x) + 36 0^{\circ} n : x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

x - 1 5^{\circ} = 9 0^{\circ} - (2 0^{\circ} + x) + 36 0^{\circ} n

Expand

9 0^{\circ} - (2 0^{\circ} + x) + 36 0^{\circ} n : - x + 36 0^{\circ} n + 7 0^{\circ}

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

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

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

Distribute parentheses

= - (2 0^{\circ}) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 2 0^{\circ} - x

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

Simplify

9 0^{\circ} - 2 0^{\circ} - x + 36 0^{\circ} n : - x + 36 0^{\circ} n + 7 0^{\circ}

9 0^{\circ} - 2 0^{\circ} - x + 36 0^{\circ} n

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

2 0^{\circ} :

multiply the denominator and numerator by

2

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

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

Add similar elements:

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

= - x + 36 0^{\circ} n + 7 0^{\circ}

= - x + 36 0^{\circ} n + 7 0^{\circ}

x - 1 5^{\circ} = - x + 36 0^{\circ} n + 7 0^{\circ}

Move

1 5^{\circ}

to the right side

x - 1 5^{\circ} = - x + 36 0^{\circ} n + 7 0^{\circ}

Add

1 5^{\circ}

to both sides

x - 1 5^{\circ} + 1 5^{\circ} = - x + 36 0^{\circ} n + 7 0^{\circ} + 1 5^{\circ}

Simplify

x - 1 5^{\circ} + 1 5^{\circ} = - x + 36 0^{\circ} n + 7 0^{\circ} + 1 5^{\circ}

Simplify

x - 1 5^{\circ} + 1 5^{\circ} : x

x - 1 5^{\circ} + 1 5^{\circ}

Add similar elements:

- 1 5^{\circ} + 1 5^{\circ} = 0

= x

Simplify

- x + 36 0^{\circ} n + 7 0^{\circ} + 1 5^{\circ} : - x + 36 0^{\circ} n + 8 5^{\circ}

- x + 36 0^{\circ} n + 7 0^{\circ} + 1 5^{\circ}

Least Common Multiplier of

18, 12 : 36

18, 12

Least Common Multiplier (LCM)

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

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

18

12

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

7 0^{\circ} :

multiply the denominator and numerator by

2

7 0^{\circ} = \frac{126 0 ^{\circ} 2}{18 \cdot 2} = 7 0^{\circ}

For

1 5^{\circ} :

multiply the denominator and numerator by

3

1 5^{\circ} = \frac{18 0 ^{\circ} 3}{12 \cdot 3} = 1 5^{\circ}

= 7 0^{\circ} + 1 5^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{252 0 ^{\circ} + 18 0 ^{\circ} 3}{36}

Add similar elements:

252 0^{\circ} + 54 0^{\circ} = 306 0^{\circ}

= - x + 36 0^{\circ} n + 8 5^{\circ}

x = - x + 36 0^{\circ} n + 8 5^{\circ}

x = - x + 36 0^{\circ} n + 8 5^{\circ}

x = - x + 36 0^{\circ} n + 8 5^{\circ}

Move

x

to the left side

x = - x + 36 0^{\circ} n + 8 5^{\circ}

Add

x

to both sides

x + x = - x + 36 0^{\circ} n + 8 5^{\circ} + x

Simplify

2 x = 36 0^{\circ} n + 8 5^{\circ}

2 x = 36 0^{\circ} n + 8 5^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} n + 8 5^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{8 5 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{36 0 ^{\circ} n}{2} + \frac{8 5 ^{\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{8 5 ^{\circ}}{2} : \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

\frac{36 0 ^{\circ} n}{2} + \frac{8 5 ^{\circ}}{2}

Apply rule

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

= \frac{36 0 ^{\circ} n + 8 5 ^{\circ}}{2}

Join

36 0^{\circ} n + 8 5^{\circ} : \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{36}

36 0^{\circ} n + 8 5^{\circ}

Convert element to fraction:

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

= \frac{36 0 ^{\circ} n \cdot 36}{36} + 8 5^{\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 36 + 306 0 ^{\circ}}{36}

Multiply the numbers:

2 \cdot 36 = 72

= \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{36}

= \frac{\frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{36}}{2}

Apply the fraction rule:

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

= \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{36 \cdot 2}

Multiply the numbers:

36 \cdot 2 = 72

= \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

x - 1 5^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 0^{\circ} + x)) + 36 0^{\circ} n :

True for all

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

x - 1 5^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 0^{\circ} + x)) + 36 0^{\circ} n

Expand

18 0^{\circ} - (9 0^{\circ} - (2 0^{\circ} + x)) + 36 0^{\circ} n : 18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n

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

Expand

9 0^{\circ} - (2 0^{\circ} + x) : - x + 7 0^{\circ}

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

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

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

Distribute parentheses

= - (2 0^{\circ}) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 2 0^{\circ} - x

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

Simplify

9 0^{\circ} - 2 0^{\circ} - x : - x + 7 0^{\circ}

9 0^{\circ} - 2 0^{\circ} - x

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

2 0^{\circ} :

multiply the denominator and numerator by

2

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

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

Add similar elements:

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

= - x + 7 0^{\circ}

= - x + 7 0^{\circ}

= 18 0^{\circ} - (- x + 7 0^{\circ}) + 36 0^{\circ} n

- (- x + 7 0^{\circ}) : x - 7 0^{\circ}

- (- x + 7 0^{\circ})

Distribute parentheses

= - (- x) - (7 0^{\circ})

Apply minus-plus rules

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

= x - 7 0^{\circ}

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

x - 1 5^{\circ} = 18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n

Move

1 5^{\circ}

to the right side

x - 1 5^{\circ} = 18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n

Add

1 5^{\circ}

to both sides

x - 1 5^{\circ} + 1 5^{\circ} = 18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n + 1 5^{\circ}

Simplify

x - 1 5^{\circ} + 1 5^{\circ} = 18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n + 1 5^{\circ}

Simplify

x - 1 5^{\circ} + 1 5^{\circ} : x

x - 1 5^{\circ} + 1 5^{\circ}

Add similar elements:

- 1 5^{\circ} + 1 5^{\circ} = 0

= x

Simplify

18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n + 1 5^{\circ} : x + 18 0^{\circ} + 36 0^{\circ} n - 5 5^{\circ}

18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n + 1 5^{\circ}

Group like terms

= x + 18 0^{\circ} + 36 0^{\circ} n + 1 5^{\circ} - 7 0^{\circ}

Least Common Multiplier of

12, 18 : 36

12, 18

Least Common Multiplier (LCM)

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

12

18

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

1 5^{\circ} :

multiply the denominator and numerator by

3

1 5^{\circ} = \frac{18 0 ^{\circ} 3}{12 \cdot 3} = 1 5^{\circ}

For

7 0^{\circ} :

multiply the denominator and numerator by

2

7 0^{\circ} = \frac{126 0 ^{\circ} 2}{18 \cdot 2} = 7 0^{\circ}

= 1 5^{\circ} - 7 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 - 252 0 ^{\circ}}{36}

Add similar elements:

54 0^{\circ} - 252 0^{\circ} = - 198 0^{\circ}

= \frac{- 198 0 ^{\circ}}{36}

Apply the fraction rule:

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

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

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

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

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

Move

x

to the left side

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

Subtract

x

from both sides

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

Simplify

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

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

Both sides are equal

True for all x; 0 = 18 0^{\circ} + 36 0^{\circ} n - 5 5^{\circ}

Since the equation is undefined for:

True for all

x

x = \frac{1296 0 ^{\circ} n + 306 0 ^{\circ}}{72}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-15)=cos(20+x) ?
The general solution for sin(x-15)=cos(20+x) is x=(12960n+3060)/(72)