What is the general solution for cos(60-x)=sin(70-3x) ?

The general solution for cos(60-x)=sin(70-3x) is x=-(2160n+180-420)/(24),x=-(900+2160n-420)/(12)

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cos(60-x)=sin(70-3x)

Solution

cos (6 0^{\circ} - x) = sin (70 - 3 x)

Solution

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}, x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

Radians

x = - \frac{π}{24} + \frac{35}{2} - \frac{12 π}{24} n, x = 35 - \frac{5 π}{12} - \frac{12 π}{12} n

Solution steps

cos (6 0^{\circ} - x) = sin (70 - 3 x)

Rewrite using trig identities

cos (6 0^{\circ} - x) = sin (70 - 3 x)

Use the following identity:

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

cos (6 0^{\circ} - x) = sin (9 0^{\circ} - (6 0^{\circ} - x))

cos (6 0^{\circ} - x) = sin (9 0^{\circ} - (6 0^{\circ} - x))

Apply trig inverse properties

cos (6 0^{\circ} - x) = sin (9 0^{\circ} - (6 0^{\circ} - x))

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

70 - 3 x = 9 0^{\circ} - (6 0^{\circ} - x) + 36 0^{\circ} n, 70 - 3 x = 18 0^{\circ} - (9 0^{\circ} - (6 0^{\circ} - x)) + 36 0^{\circ} n

70 - 3 x = 9 0^{\circ} - (6 0^{\circ} - x) + 36 0^{\circ} n, 70 - 3 x = 18 0^{\circ} - (9 0^{\circ} - (6 0^{\circ} - x)) + 36 0^{\circ} n

70 - 3 x = 9 0^{\circ} - (6 0^{\circ} - x) + 36 0^{\circ} n : x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

70 - 3 x = 9 0^{\circ} - (6 0^{\circ} - x) + 36 0^{\circ} n

Expand

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

9 0^{\circ} - (6 0^{\circ} - x) + 36 0^{\circ} n

- (6 0^{\circ} - x) : - 6 0^{\circ} + x

- (6 0^{\circ} - x)

Distribute parentheses

= - (6 0^{\circ}) - (- x)

Apply minus-plus rules

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

= - 6 0^{\circ} + x

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

Simplify

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

9 0^{\circ} - 6 0^{\circ} + x + 36 0^{\circ} n

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 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}

For

6 0^{\circ} :

multiply the denominator and numerator by

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 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 - 18 0 ^{\circ} 2}{6}

Add similar elements:

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

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

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

70 - 3 x = x + 36 0^{\circ} n + 3 0^{\circ}

Move

70

to the right side

70 - 3 x = x + 36 0^{\circ} n + 3 0^{\circ}

Subtract

70

from both sides

70 - 3 x - 70 = x + 36 0^{\circ} n + 3 0^{\circ} - 70

Simplify

- 3 x = x + 36 0^{\circ} n + 3 0^{\circ} - 70

- 3 x = x + 36 0^{\circ} n + 3 0^{\circ} - 70

Move

x

to the left side

- 3 x = x + 36 0^{\circ} n + 3 0^{\circ} - 70

Subtract

x

from both sides

- 3 x - x = x + 36 0^{\circ} n + 3 0^{\circ} - 70 - x

Simplify

- 4 x = 36 0^{\circ} n + 3 0^{\circ} - 70

- 4 x = 36 0^{\circ} n + 3 0^{\circ} - 70

Divide both sides by

- 4

- 4 x = 36 0^{\circ} n + 3 0^{\circ} - 70

Divide both sides by

- 4

\frac{- 4 x}{- 4} = \frac{36 0 ^{\circ} n}{- 4} + \frac{3 0 ^{\circ}}{- 4} - \frac{70}{- 4}

Simplify

\frac{- 4 x}{- 4} = \frac{36 0 ^{\circ} n}{- 4} + \frac{3 0 ^{\circ}}{- 4} - \frac{70}{- 4}

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{- 4} + \frac{3 0 ^{\circ}}{- 4} - \frac{70}{- 4} : - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

\frac{36 0 ^{\circ} n}{- 4} + \frac{3 0 ^{\circ}}{- 4} - \frac{70}{- 4}

Apply rule

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

= \frac{36 0 ^{\circ} n + 3 0 ^{\circ} - 70}{- 4}

Apply the fraction rule:

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

= - \frac{36 0 ^{\circ} n + 3 0 ^{\circ} - 70}{4}

Join

36 0^{\circ} n + 3 0^{\circ} - 70 : \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6}

36 0^{\circ} n + 3 0^{\circ} - 70

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 6}{6}, 70 = \frac{70 \cdot 6}{6}

= \frac{36 0 ^{\circ} n \cdot 6}{6} + 3 0^{\circ} - \frac{70 \cdot 6}{6}

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 6 + 18 0 ^{\circ} - 70 \cdot 6}{6}

36 0^{\circ} n \cdot 6 + 18 0^{\circ} - 70 \cdot 6 = 216 0^{\circ} n + 18 0^{\circ} - 420

36 0^{\circ} n \cdot 6 + 18 0^{\circ} - 70 \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 216 0^{\circ} n + 18 0^{\circ} - 70 \cdot 6

Multiply the numbers:

70 \cdot 6 = 420

= 216 0^{\circ} n + 18 0^{\circ} - 420

= \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6}

= - \frac{\frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6}}{4}

Simplify

\frac{\frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6}}{4} : \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

\frac{\frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6}}{4}

Apply the fraction rule:

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

= \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

= - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}

70 - 3 x = 18 0^{\circ} - (9 0^{\circ} - (6 0^{\circ} - x)) + 36 0^{\circ} n : x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

70 - 3 x = 18 0^{\circ} - (9 0^{\circ} - (6 0^{\circ} - x)) + 36 0^{\circ} n

Expand

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

18 0^{\circ} - (9 0^{\circ} - (6 0^{\circ} - x)) + 36 0^{\circ} n

Expand

9 0^{\circ} - (6 0^{\circ} - x) : x + 3 0^{\circ}

9 0^{\circ} - (6 0^{\circ} - x)

- (6 0^{\circ} - x) : - 6 0^{\circ} + x

- (6 0^{\circ} - x)

Distribute parentheses

= - (6 0^{\circ}) - (- x)

Apply minus-plus rules

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

= - 6 0^{\circ} + x

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

Simplify

9 0^{\circ} - 6 0^{\circ} + x : x + 3 0^{\circ}

9 0^{\circ} - 6 0^{\circ} + x

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 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}

For

6 0^{\circ} :

multiply the denominator and numerator by

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 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 - 18 0 ^{\circ} 2}{6}

Add similar elements:

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

= x + 3 0^{\circ}

= x + 3 0^{\circ}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - x - 3 0^{\circ}

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

70 - 3 x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n

Move

70

to the right side

70 - 3 x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n

Subtract

70

from both sides

70 - 3 x - 70 = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n - 70

Simplify

- 3 x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n - 70

- 3 x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n - 70

Move

x

to the left side

- 3 x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n - 70

Add

x

to both sides

- 3 x + x = 18 0^{\circ} - x - 3 0^{\circ} + 36 0^{\circ} n - 70 + x

Simplify

- 2 x = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n - 70

- 2 x = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n - 70

Divide both sides by

- 2

- 2 x = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n - 70

Divide both sides by

- 2

\frac{- 2 x}{- 2} = \frac{18 0 ^{\circ}}{- 2} - \frac{3 0 ^{\circ}}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{70}{- 2}

Simplify

\frac{- 2 x}{- 2} = \frac{18 0 ^{\circ}}{- 2} - \frac{3 0 ^{\circ}}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{70}{- 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{3 0 ^{\circ}}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{70}{- 2} : - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

\frac{18 0 ^{\circ}}{- 2} - \frac{3 0 ^{\circ}}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{70}{- 2}

Group like terms

= \frac{18 0 ^{\circ}}{- 2} - \frac{70}{- 2} + \frac{36 0 ^{\circ} n}{- 2} - \frac{3 0 ^{\circ}}{- 2}

Apply rule

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

= \frac{18 0 ^{\circ} - 70 + 36 0 ^{\circ} n - 3 0 ^{\circ}}{- 2}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} - 70 + 36 0 ^{\circ} n - 3 0 ^{\circ}}{2}

Join

18 0^{\circ} - 70 + 36 0^{\circ} n - 3 0^{\circ} : \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6}

18 0^{\circ} - 70 + 36 0^{\circ} n - 3 0^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 70 = \frac{70 \cdot 6}{6}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 6}{6}

= 18 0^{\circ} - \frac{70 \cdot 6}{6} + \frac{36 0 ^{\circ} n \cdot 6}{6} - 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} 6 - 70 \cdot 6 + 36 0 ^{\circ} n \cdot 6 - 18 0 ^{\circ}}{6}

18 0^{\circ} 6 - 70 \cdot 6 + 36 0^{\circ} n \cdot 6 - 18 0^{\circ} = 90 0^{\circ} + 216 0^{\circ} n - 420

18 0^{\circ} 6 - 70 \cdot 6 + 36 0^{\circ} n \cdot 6 - 18 0^{\circ}

Group like terms

= 108 0^{\circ} - 18 0^{\circ} + 2 \cdot 108 0^{\circ} n - 70 \cdot 6

Add similar elements:

108 0^{\circ} - 18 0^{\circ} = 90 0^{\circ}

= 90 0^{\circ} + 2 \cdot 108 0^{\circ} n - 70 \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 90 0^{\circ} + 216 0^{\circ} n - 70 \cdot 6

Multiply the numbers:

70 \cdot 6 = 420

= 90 0^{\circ} + 216 0^{\circ} n - 420

= \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6}

= - \frac{\frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6}}{2}

Simplify

\frac{\frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6}}{2} : \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

\frac{\frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6}}{2}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

= - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}, x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

x = - \frac{216 0 ^{\circ} n + 18 0 ^{\circ} - 420}{24}, x = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n - 420}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(60-x)=sin(70-3x) ?
The general solution for cos(60-x)=sin(70-3x) is x=-(2160n+180-420)/(24),x=-(900+2160n-420)/(12)