What is the general solution for cos(3x)=sin(x-30) ?

The general solution for cos(3x)=sin(x-30) is x=(180(3n+1))/6 ,x=-(180(3n+1))/3

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cos(3x)=sin(x-30)

Solution

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

Solution

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}, x = - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

Radians

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

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

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

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

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

Apply trig inverse properties

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

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

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

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

x - 3 0^{\circ} = 9 0^{\circ} - 3 x + 36 0^{\circ} n : x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

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

Move

3 0^{\circ}

to the right side

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

Add

3 0^{\circ}

to both sides

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

Simplify

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

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

Group like terms

= - 3 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

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

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

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

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

Move

3 x

to the left side

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

Add

3 x

to both sides

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

Simplify

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

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

Divide both sides by

4

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

Divide both sides by

4

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{12 0 ^{\circ}}{4}

Simplify

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{12 0 ^{\circ}}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{4} + \frac{12 0 ^{\circ}}{4} : \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

\frac{36 0 ^{\circ} n}{4} + \frac{12 0 ^{\circ}}{4}

Apply rule

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

= \frac{36 0 ^{\circ} n + 12 0 ^{\circ}}{4}

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}}{4}

Apply the fraction rule:

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

= \frac{108 0 ^{\circ} n + 36 0 ^{\circ}}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

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

Factor

108 0^{\circ} n + 36 0^{\circ} : 36 0^{\circ} (3 n + 1)

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

Rewrite as

= 3 \cdot 36 0^{\circ} n + 1 \cdot 36 0^{\circ}

Factor out common term

36 0^{\circ}

= 36 0^{\circ} (3 n + 1)

= \frac{36 0 ^{\circ} ( 3 n + 1 )}{12}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}

x - 3 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - 3 x) + 36 0^{\circ} n : x = - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

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

Expand

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

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

- (9 0^{\circ} - 3 x) : - 9 0^{\circ} + 3 x

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

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

Move

3 0^{\circ}

to the right side

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

Simplify

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

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

Group like terms

= 3 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

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

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

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

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

Move

3 x

to the left side

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

Subtract

3 x

from both sides

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

Simplify

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

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

Divide both sides by

- 2

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

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{18 0 ^{\circ}}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{6 0 ^{\circ}}{- 2} : - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

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

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}}{- 2}

Apply the fraction rule:

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

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

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}}{2}

Simplify

\frac{\frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}}{2} : \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{6}

\frac{\frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3}}{2}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{6}

= - \frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{6}

Cancel

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

\frac{36 0 ^{\circ} + 108 0 ^{\circ} n}{6}

Factor

36 0^{\circ} + 108 0^{\circ} n : 36 0^{\circ} (1 + 3 n)

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

Rewrite as

= 1 \cdot 36 0^{\circ} + 3 \cdot 36 0^{\circ} n

Factor out common term

36 0^{\circ}

= 36 0^{\circ} (1 + 3 n)

= \frac{36 0 ^{\circ} ( 1 + 3 n )}{6}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

= - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

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

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

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

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}, x = - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

x = \frac{18 0 ^{\circ} ( 3 n + 1 )}{6}, x = - \frac{18 0 ^{\circ} ( 3 n + 1 )}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(3x)=sin(x-30) ?
The general solution for cos(3x)=sin(x-30) is x=(180(3n+1))/6 ,x=-(180(3n+1))/3