What is the general solution for sin(x-30)=cos(2x) ?

The general solution for sin(x-30)=cos(2x) is x=(1080n+360)/9 ,x=-(360+1080n)/3

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Popular Trigonometry >

sin(x-30)=cos(2x)

Solution

sin (x - 3 0^{\circ}) = cos (2 x)

Solution

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}, x = - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}

Radians

x = \frac{2 π}{9} + \frac{6 π}{9} n, x = - \frac{2 π}{3} - \frac{6 π}{3} n

Solution steps

sin (x - 3 0^{\circ}) = cos (2 x)

Rewrite using trig identities

sin (x - 3 0^{\circ}) = cos (2 x)

Use the following identity:

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

sin (x - 3 0^{\circ}) = sin (9 0^{\circ} - 2 x)

sin (x - 3 0^{\circ}) = sin (9 0^{\circ} - 2 x)

Apply trig inverse properties

sin (x - 3 0^{\circ}) = sin (9 0^{\circ} - 2 x)

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

x - 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n, x - 3 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

x - 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n, x - 3 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

x - 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n : x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

x - 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

x - 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n

Add

3 0^{\circ}

to both sides

x - 3 0^{\circ} + 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n + 3 0^{\circ}

Simplify

x - 3 0^{\circ} + 3 0^{\circ} = 9 0^{\circ} - 2 x + 36 0^{\circ} n + 3 0^{\circ}

Simplify

x - 3 0^{\circ} + 3 0^{\circ} : x

x - 3 0^{\circ} + 3 0^{\circ}

Add similar elements:

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

= x

Simplify

9 0^{\circ} - 2 x + 36 0^{\circ} n + 3 0^{\circ} : - 2 x + 36 0^{\circ} n + 12 0^{\circ}

9 0^{\circ} - 2 x + 36 0^{\circ} n + 3 0^{\circ}

Group like terms

= - 2 x + 36 0^{\circ} n + 9 0^{\circ} + 3 0^{\circ}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

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

2

6

= 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}

= 9 0^{\circ} + 3 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}}{6}

Add similar elements:

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

= 12 0^{\circ}

Cancel the common factor:

2

= - 2 x + 36 0^{\circ} n + 12 0^{\circ}

x = - 2 x + 36 0^{\circ} n + 12 0^{\circ}

x = - 2 x + 36 0^{\circ} n + 12 0^{\circ}

x = - 2 x + 36 0^{\circ} n + 12 0^{\circ}

Move

2 x

to the left side

x = - 2 x + 36 0^{\circ} n + 12 0^{\circ}

Add

2 x

to both sides

x + 2 x = - 2 x + 36 0^{\circ} n + 12 0^{\circ} + 2 x

Simplify

3 x = 36 0^{\circ} n + 12 0^{\circ}

3 x = 36 0^{\circ} n + 12 0^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 12 0^{\circ}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{12 0 ^{\circ}}{3} : \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

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

Apply rule

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

= \frac{36 0 ^{\circ} n + 12 0 ^{\circ}}{3}

Join

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

36 0^{\circ} n + 12 0^{\circ}

Convert element to fraction:

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

= \frac{36 0 ^{\circ} n \cdot 3}{3} + 12 0^{\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 3 + 36 0 ^{\circ}}{3}

Multiply the numbers:

2 \cdot 3 = 6

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

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

Apply the fraction rule:

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

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

Multiply the numbers:

3 \cdot 3 = 9

= \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}

x - 3 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n : x = - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}

x - 3 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - 2 x) + 36 0^{\circ} n

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

x - 3 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

x - 3 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n

Add

3 0^{\circ}

to both sides

x - 3 0^{\circ} + 3 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n + 3 0^{\circ}

Simplify

x - 3 0^{\circ} + 3 0^{\circ} = 18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n + 3 0^{\circ}

Simplify

x - 3 0^{\circ} + 3 0^{\circ} : x

x - 3 0^{\circ} + 3 0^{\circ}

Add similar elements:

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

= x

Simplify

18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n + 3 0^{\circ} : 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

18 0^{\circ} - 9 0^{\circ} + 2 x + 36 0^{\circ} n + 3 0^{\circ}

Group like terms

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 9 0^{\circ} + 3 0^{\circ}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

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

2

6

= 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}

= - 9 0^{\circ} + 3 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}}{6}

Add similar elements:

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

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

Apply the fraction rule:

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

= - 6 0^{\circ}

Cancel the common factor:

2

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Move

2 x

to the left side

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Subtract

2 x

from both sides

x - 2 x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ} - 2 x

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 0 ^{\circ}}{- 1} : - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{6 0 ^{\circ}}{- 1}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 6 0 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 6 0 ^{\circ}}{1}

Join

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

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

Convert element to fraction:

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

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

18 0^{\circ} 3 + 36 0^{\circ} n \cdot 3 - 18 0^{\circ} = 36 0^{\circ} + 108 0^{\circ} n

18 0^{\circ} 3 + 36 0^{\circ} n \cdot 3 - 18 0^{\circ}

Add similar elements:

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

= 36 0^{\circ} + 2 \cdot 54 0^{\circ} n

Multiply the numbers:

2 \cdot 3 = 6

= 36 0^{\circ} + 108 0^{\circ} n

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

= - \frac{\frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

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

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

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

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}, x = - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}

x = \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{9}, x = - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-30)=cos(2x) ?
The general solution for sin(x-30)=cos(2x) is x=(1080n+360)/9 ,x=-(360+1080n)/3