What is the general solution for sin(3x+10)=cos(x+20) ?

The general solution for sin(3x+10)=cos(x+20) is x=(1080n+180)/(12),x=(900+3240n)/(18)

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sin(3x+10)=cos(x+20)

Solution

sin (3 x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

Solution

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

Radians

x = \frac{π}{12} + \frac{6 π}{12} n, x = \frac{5 π}{18} + \frac{18 π}{18} n

Solution steps

sin (3 x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

Rewrite using trig identities

sin (3 x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

Use the following identity:

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

sin (3 x + 1 0^{\circ}) = sin (9 0^{\circ} - (x + 2 0^{\circ}))

sin (3 x + 1 0^{\circ}) = sin (9 0^{\circ} - (x + 2 0^{\circ}))

Apply trig inverse properties

sin (3 x + 1 0^{\circ}) = sin (9 0^{\circ} - (x + 2 0^{\circ}))

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

3 x + 1 0^{\circ} = 9 0^{\circ} - (x + 2 0^{\circ}) + 36 0^{\circ} n, 3 x + 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 2 0^{\circ})) + 36 0^{\circ} n

3 x + 1 0^{\circ} = 9 0^{\circ} - (x + 2 0^{\circ}) + 36 0^{\circ} n, 3 x + 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 2 0^{\circ})) + 36 0^{\circ} n

3 x + 1 0^{\circ} = 9 0^{\circ} - (x + 2 0^{\circ}) + 36 0^{\circ} n : x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

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

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - x - 2 0^{\circ}

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

Simplify

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

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

Group like terms

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

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}

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

Move

1 0^{\circ}

to the right side

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

Subtract

1 0^{\circ}

from both sides

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

Simplify

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

Simplify

3 x + 1 0^{\circ} - 1 0^{\circ} : 3 x

3 x + 1 0^{\circ} - 1 0^{\circ}

Add similar elements:

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

= 3 x

Simplify

- x + 36 0^{\circ} n + 7 0^{\circ} - 1 0^{\circ} : - x + 36 0^{\circ} n + 6 0^{\circ}

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

Combine the fractions

7 0^{\circ} - 1 0^{\circ} : 6 0^{\circ}

Apply rule

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

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

Add similar elements:

126 0^{\circ} - 18 0^{\circ} = 108 0^{\circ}

= 6 0^{\circ}

Cancel the common factor:

6

= 6 0^{\circ}

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

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

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

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

Move

x

to the left side

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

Add

x

to both sides

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

Simplify

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

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

Divide both sides by

4

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

Divide both sides by

4

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{6 0 ^{\circ}}{4}

Simplify

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{6 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{6 0 ^{\circ}}{4} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

\frac{36 0 ^{\circ} n}{4} + \frac{6 0 ^{\circ}}{4}

Apply rule

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

= \frac{36 0 ^{\circ} n + 6 0 ^{\circ}}{4}

Join

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}

= \frac{\frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}}{4}

Apply the fraction rule:

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

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}

3 x + 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 2 0^{\circ})) + 36 0^{\circ} n : x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

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

Expand

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

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

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - x - 2 0^{\circ}

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

Simplify

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

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

Group like terms

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

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

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

Move

1 0^{\circ}

to the right side

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

Subtract

1 0^{\circ}

from both sides

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

Simplify

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

Simplify

3 x + 1 0^{\circ} - 1 0^{\circ} : 3 x

3 x + 1 0^{\circ} - 1 0^{\circ}

Add similar elements:

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

= 3 x

Simplify

18 0^{\circ} + x - 7 0^{\circ} + 36 0^{\circ} n - 1 0^{\circ} : x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

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

Group like terms

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

Combine the fractions

- 1 0^{\circ} - 7 0^{\circ} : - 8 0^{\circ}

Apply rule

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

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

Add similar elements:

- 18 0^{\circ} - 126 0^{\circ} = - 144 0^{\circ}

= \frac{- 144 0 ^{\circ}}{18}

Apply the fraction rule:

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

= - 8 0^{\circ}

Cancel the common factor:

2

= - 8 0^{\circ}

= x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

3 x = x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

3 x = x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

3 x = x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

Move

x

to the left side

3 x = x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

Subtract

x

from both sides

3 x - x = x + 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ} - x

Simplify

2 x = 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

2 x = 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

Divide both sides by

2

2 x = 18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} - \frac{8 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = 9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} - \frac{8 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} - \frac{8 0 ^{\circ}}{2} : \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} - \frac{8 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 - 8 0 ^{\circ}}{2}

Join

18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ} : \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{9}

18 0^{\circ} + 36 0^{\circ} n - 8 0^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 9}{9}

= 18 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 9}{9} - 8 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 + 36 0 ^{\circ} n \cdot 9 - 72 0 ^{\circ}}{9}

18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9 - 72 0^{\circ} = 90 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} 9 + 36 0^{\circ} n \cdot 9 - 72 0^{\circ}

Add similar elements:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= 90 0^{\circ} + 2 \cdot 162 0^{\circ} n

Multiply the numbers:

2 \cdot 9 = 18

= 90 0^{\circ} + 324 0^{\circ} n

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{9}

= \frac{\frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{9}}{2}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{18}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x+10)=cos(x+20) ?
The general solution for sin(3x+10)=cos(x+20) is x=(1080n+180)/(12),x=(900+3240n)/(18)