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

The general solution for sin(x+20)=cos(x) is x=-360n+35,x=-145-360n

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(x+20)=cos(x)

Solution

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

Solution

x = - 36 0^{\circ} n + 3 5^{\circ}, x = - 14 5^{\circ} - 36 0^{\circ} n

Radians

x = \frac{7 π}{36} - 2 πn, x = - \frac{29 π}{36} - 2 πn

Solution steps

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

Subtract

cos (x)

from both sides

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

Rewrite using trig identities

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

Use the following identity:

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

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

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}

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

Use the Sum to Product identity:

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

= - 2 sin (\frac{7 0 ^{\circ} - x + x}{2}) sin (\frac{7 0 ^{\circ} - x - x}{2})

Simplify

- 2 sin (\frac{7 0 ^{\circ} - x + x}{2}) sin (\frac{7 0 ^{\circ} - x - x}{2}) : - 2 sin (3 5^{\circ}) sin (\frac{126 0 ^{\circ} - 36 x}{36})

- 2 sin (\frac{7 0 ^{\circ} - x + x}{2}) sin (\frac{7 0 ^{\circ} - x - x}{2})

\frac{7 0 ^{\circ} - x + x}{2} = 3 5^{\circ}

\frac{7 0 ^{\circ} - x + x}{2}

Add similar elements:

- x + x = 0

= \frac{7 0 ^{\circ}}{2}

Apply the fraction rule:

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

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

Multiply the numbers:

18 \cdot 2 = 36

= 3 5^{\circ}

= - 2 sin (3 5^{\circ}) sin (\frac{- x - x + 7 0 ^{\circ}}{2})

\frac{7 0 ^{\circ} - x - x}{2} = \frac{126 0 ^{\circ} - 36 x}{36}

\frac{7 0 ^{\circ} - x - x}{2}

Add similar elements:

- x - x = - 2 x

= \frac{7 0 ^{\circ} - 2 x}{2}

Join

7 0^{\circ} - 2 x : \frac{126 0 ^{\circ} - 36 x}{18}

7 0^{\circ} - 2 x

Convert element to fraction:

2 x = \frac{2 x 18}{18}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 18 = 36

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

= \frac{\frac{126 0 ^{\circ} - 36 x}{18}}{2}

Apply the fraction rule:

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

= \frac{126 0 ^{\circ} - 36 x}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= \frac{126 0 ^{\circ} - 36 x}{36}

= - 2 sin (3 5^{\circ}) sin (\frac{- 36 x + 126 0 ^{\circ}}{36})

= - 2 sin (3 5^{\circ}) sin (\frac{126 0 ^{\circ} - 36 x}{36})

- 2 sin (3 5^{\circ}) sin (\frac{126 0 ^{\circ} - 36 x}{36}) = 0

Divide both sides by

- 2 sin (3 5^{\circ})

- 2 sin (3 5^{\circ}) sin (\frac{126 0 ^{\circ} - 36 x}{36}) = 0

Divide both sides by

- 2 sin (3 5^{\circ})

\frac{- 2 sin ( 3 5 ^{\circ} ) sin ( \frac{126 0 ^{\circ} - 36 x}{36} )}{- 2 sin ( 3 5 ^{\circ} )} = \frac{0}{- 2 sin ( 3 5 ^{\circ} )}

Simplify

sin (\frac{126 0 ^{\circ} - 36 x}{36}) = 0

sin (\frac{126 0 ^{\circ} - 36 x}{36}) = 0

General solutions for

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

\frac{126 0 ^{\circ} - 36 x}{36} = 0 + 36 0^{\circ} n, \frac{126 0 ^{\circ} - 36 x}{36} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{126 0 ^{\circ} - 36 x}{36} = 0 + 36 0^{\circ} n : x = - 36 0^{\circ} n + 3 5^{\circ}

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

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

\frac{126 0 ^{\circ} - 36 x}{36} = 36 0^{\circ} n

Multiply both sides by

36

\frac{126 0 ^{\circ} - 36 x}{36} = 36 0^{\circ} n

Multiply both sides by

36

\frac{36 ( 126 0 ^{\circ} - 36 x )}{36} = 36 \cdot 36 0^{\circ} n

Simplify

126 0^{\circ} - 36 x = 1296 0^{\circ} n

126 0^{\circ} - 36 x = 1296 0^{\circ} n

Move

126 0^{\circ}

to the right side

126 0^{\circ} - 36 x = 1296 0^{\circ} n

Subtract

126 0^{\circ}

from both sides

126 0^{\circ} - 36 x - 126 0^{\circ} = 1296 0^{\circ} n - 126 0^{\circ}

Simplify

- 36 x = 1296 0^{\circ} n - 126 0^{\circ}

- 36 x = 1296 0^{\circ} n - 126 0^{\circ}

Divide both sides by

- 36

- 36 x = 1296 0^{\circ} n - 126 0^{\circ}

Divide both sides by

- 36

\frac{- 36 x}{- 36} = \frac{1296 0 ^{\circ} n}{- 36} - \frac{126 0 ^{\circ}}{- 36}

Simplify

\frac{- 36 x}{- 36} = \frac{1296 0 ^{\circ} n}{- 36} - \frac{126 0 ^{\circ}}{- 36}

Simplify

\frac{- 36 x}{- 36} : x

\frac{- 36 x}{- 36}

Apply the fraction rule:

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

= \frac{36 x}{36}

Divide the numbers:

\frac{36}{36} = 1

= x

Simplify

\frac{1296 0 ^{\circ} n}{- 36} - \frac{126 0 ^{\circ}}{- 36} : - 36 0^{\circ} n + 3 5^{\circ}

\frac{1296 0 ^{\circ} n}{- 36} - \frac{126 0 ^{\circ}}{- 36}

\frac{1296 0 ^{\circ} n}{- 36} = - 36 0^{\circ} n

\frac{1296 0 ^{\circ} n}{- 36}

Apply the fraction rule:

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

= - \frac{1296 0 ^{\circ} n}{36}

Divide the numbers:

\frac{72}{36} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{126 0 ^{\circ}}{- 36}

Apply the fraction rule:

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

= - 36 0^{\circ} n - (- 3 5^{\circ})

Apply rule

- (- a) = a

= - 36 0^{\circ} n + 3 5^{\circ}

x = - 36 0^{\circ} n + 3 5^{\circ}

x = - 36 0^{\circ} n + 3 5^{\circ}

x = - 36 0^{\circ} n + 3 5^{\circ}

Solve

\frac{126 0 ^{\circ} - 36 x}{36} = 18 0^{\circ} + 36 0^{\circ} n : x = - 14 5^{\circ} - 36 0^{\circ} n

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

Multiply both sides by

36

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

Multiply both sides by

36

\frac{36 ( 126 0 ^{\circ} - 36 x )}{36} = 648 0^{\circ} + 36 \cdot 36 0^{\circ} n

Simplify

126 0^{\circ} - 36 x = 648 0^{\circ} + 1296 0^{\circ} n

126 0^{\circ} - 36 x = 648 0^{\circ} + 1296 0^{\circ} n

Move

126 0^{\circ}

to the right side

126 0^{\circ} - 36 x = 648 0^{\circ} + 1296 0^{\circ} n

Subtract

126 0^{\circ}

from both sides

126 0^{\circ} - 36 x - 126 0^{\circ} = 648 0^{\circ} + 1296 0^{\circ} n - 126 0^{\circ}

Simplify

- 36 x = 522 0^{\circ} + 1296 0^{\circ} n

- 36 x = 522 0^{\circ} + 1296 0^{\circ} n

Divide both sides by

- 36

- 36 x = 522 0^{\circ} + 1296 0^{\circ} n

Divide both sides by

- 36

\frac{- 36 x}{- 36} = \frac{522 0 ^{\circ}}{- 36} + \frac{1296 0 ^{\circ} n}{- 36}

Simplify

\frac{- 36 x}{- 36} = \frac{522 0 ^{\circ}}{- 36} + \frac{1296 0 ^{\circ} n}{- 36}

Simplify

\frac{- 36 x}{- 36} : x

\frac{- 36 x}{- 36}

Apply the fraction rule:

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

= \frac{36 x}{36}

Divide the numbers:

\frac{36}{36} = 1

= x

Simplify

\frac{522 0 ^{\circ}}{- 36} + \frac{1296 0 ^{\circ} n}{- 36} : - 14 5^{\circ} - 36 0^{\circ} n

\frac{522 0 ^{\circ}}{- 36} + \frac{1296 0 ^{\circ} n}{- 36}

Apply the fraction rule:

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

= - 14 5^{\circ} + \frac{1296 0 ^{\circ} n}{- 36}

\frac{1296 0 ^{\circ} n}{- 36} = - 36 0^{\circ} n

\frac{1296 0 ^{\circ} n}{- 36}

Apply the fraction rule:

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

= - \frac{1296 0 ^{\circ} n}{36}

Divide the numbers:

\frac{72}{36} = 2

= - 36 0^{\circ} n

= - 14 5^{\circ} - 36 0^{\circ} n

x = - 14 5^{\circ} - 36 0^{\circ} n

x = - 14 5^{\circ} - 36 0^{\circ} n

x = - 14 5^{\circ} - 36 0^{\circ} n

x = - 36 0^{\circ} n + 3 5^{\circ}, x = - 14 5^{\circ} - 36 0^{\circ} n

Popular Examples

cos(x)-sqrt(3)sin(x)=1

cos (x) - 3 sin (x) = 1

1-sin(x)=cos(x)

1 - sin (x) = cos (x)

cos^2(θ)-cos(θ)=0

cos^{2} (θ) - cos (θ) = 0

cos(θ)= 12/13

cos (θ) = \frac{12}{13}

sin(2θ)=cos(2θ)

sin (2 θ) = cos (2 θ)

Frequently Asked Questions (FAQ)

What is the general solution for sin(x+20)=cos(x) ?
The general solution for sin(x+20)=cos(x) is x=-360n+35,x=-145-360n