What is the general solution for sin(20-θ)-cos(10+3θ)=0 ?

The general solution for sin(20-θ)-cos(10+3θ)=0 is θ=(1080n+180)/6 ,θ=-(180(9n+2))/(18)

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sin(20-θ)-cos(10+3θ)=0

Solution

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

Solution

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}, θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

Radians

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

Solution steps

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

Add

cos (1 0^{\circ} + 3 θ)

to both sides

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

Rewrite using trig identities

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

Use the following identity:

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

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

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

Apply trig inverse properties

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

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

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

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

2 0^{\circ} - θ = 9 0^{\circ} - (1 0^{\circ} + 3 θ) + 36 0^{\circ} n : θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

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

Expand

9 0^{\circ} - (1 0^{\circ} + 3 θ) + 36 0^{\circ} n : - 3 θ + 36 0^{\circ} n + 8 0^{\circ}

9 0^{\circ} - (1 0^{\circ} + 3 θ) + 36 0^{\circ} n

- (1 0^{\circ} + 3 θ) : - 1 0^{\circ} - 3 θ

- (1 0^{\circ} + 3 θ)

Distribute parentheses

= - (1 0^{\circ}) - (3 θ)

Apply minus-plus rules

+ (- a) = - a

= - 1 0^{\circ} - 3 θ

= 9 0^{\circ} - 1 0^{\circ} - 3 θ + 36 0^{\circ} n

Simplify

9 0^{\circ} - 1 0^{\circ} - 3 θ + 36 0^{\circ} n : - 3 θ + 36 0^{\circ} n + 8 0^{\circ}

9 0^{\circ} - 1 0^{\circ} - 3 θ + 36 0^{\circ} n

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

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

2

18

= 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}

= 9 0^{\circ} - 1 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}}{18}

Add similar elements:

162 0^{\circ} - 18 0^{\circ} = 144 0^{\circ}

= 8 0^{\circ}

Cancel the common factor:

2

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

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

2 0^{\circ} - θ = - 3 θ + 36 0^{\circ} n + 8 0^{\circ}

Move

2 0^{\circ}

to the right side

2 0^{\circ} - θ = - 3 θ + 36 0^{\circ} n + 8 0^{\circ}

Subtract

2 0^{\circ}

from both sides

2 0^{\circ} - θ - 2 0^{\circ} = - 3 θ + 36 0^{\circ} n + 8 0^{\circ} - 2 0^{\circ}

Simplify

2 0^{\circ} - θ - 2 0^{\circ} = - 3 θ + 36 0^{\circ} n + 8 0^{\circ} - 2 0^{\circ}

Simplify

2 0^{\circ} - θ - 2 0^{\circ} : - θ

2 0^{\circ} - θ - 2 0^{\circ}

Add similar elements:

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

= - θ

Simplify

- 3 θ + 36 0^{\circ} n + 8 0^{\circ} - 2 0^{\circ} : - 3 θ + 36 0^{\circ} n + 6 0^{\circ}

- 3 θ + 36 0^{\circ} n + 8 0^{\circ} - 2 0^{\circ}

Combine the fractions

8 0^{\circ} - 2 0^{\circ} : 6 0^{\circ}

Apply rule

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

= \frac{72 0 ^{\circ} - 18 0 ^{\circ}}{9}

Add similar elements:

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

= 6 0^{\circ}

Cancel the common factor:

3

= 6 0^{\circ}

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

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

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

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

Move

3 θ

to the left side

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

Add

3 θ

to both sides

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

Simplify

2 θ = 36 0^{\circ} n + 6 0^{\circ}

2 θ = 36 0^{\circ} n + 6 0^{\circ}

Divide both sides by

2

2 θ = 36 0^{\circ} n + 6 0^{\circ}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{6 0 ^{\circ}}{2} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

\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{36 0 ^{\circ} n + 6 0 ^{\circ}}{2}

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}}{2}

Apply the fraction rule:

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

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}

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

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

Expand

18 0^{\circ} - (9 0^{\circ} - (1 0^{\circ} + 3 θ)) + 36 0^{\circ} n : 18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n

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

Expand

9 0^{\circ} - (1 0^{\circ} + 3 θ) : - 3 θ + 8 0^{\circ}

9 0^{\circ} - (1 0^{\circ} + 3 θ)

- (1 0^{\circ} + 3 θ) : - 1 0^{\circ} - 3 θ

- (1 0^{\circ} + 3 θ)

Distribute parentheses

= - (1 0^{\circ}) - (3 θ)

Apply minus-plus rules

+ (- a) = - a

= - 1 0^{\circ} - 3 θ

= 9 0^{\circ} - 1 0^{\circ} - 3 θ

Simplify

9 0^{\circ} - 1 0^{\circ} - 3 θ : - 3 θ + 8 0^{\circ}

9 0^{\circ} - 1 0^{\circ} - 3 θ

Least Common Multiplier of

2, 18 : 18

2, 18

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

18 : 2 \cdot 3 \cdot 3

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

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

2

18

= 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}

= 9 0^{\circ} - 1 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}}{18}

Add similar elements:

162 0^{\circ} - 18 0^{\circ} = 144 0^{\circ}

= 8 0^{\circ}

Cancel the common factor:

2

= - 3 θ + 8 0^{\circ}

= - 3 θ + 8 0^{\circ}

= 18 0^{\circ} - (- 3 θ + 8 0^{\circ}) + 36 0^{\circ} n

- (- 3 θ + 8 0^{\circ}) : 3 θ - 8 0^{\circ}

- (- 3 θ + 8 0^{\circ})

Distribute parentheses

= - (- 3 θ) - (8 0^{\circ})

Apply minus-plus rules

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

= 3 θ - 8 0^{\circ}

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

2 0^{\circ} - θ = 18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

2 0^{\circ} - θ = 18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n

Subtract

2 0^{\circ}

from both sides

2 0^{\circ} - θ - 2 0^{\circ} = 18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Simplify

2 0^{\circ} - θ - 2 0^{\circ} = 18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Simplify

2 0^{\circ} - θ - 2 0^{\circ} : - θ

2 0^{\circ} - θ - 2 0^{\circ}

Add similar elements:

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

= - θ

Simplify

18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ} : 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

18 0^{\circ} + 3 θ - 8 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

Group like terms

= 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ} - 8 0^{\circ}

Combine the fractions

- 2 0^{\circ} - 8 0^{\circ} : - 10 0^{\circ}

Apply rule

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

= \frac{- 18 0 ^{\circ} - 72 0 ^{\circ}}{9}

Add similar elements:

- 18 0^{\circ} - 72 0^{\circ} = - 90 0^{\circ}

= \frac{- 90 0 ^{\circ}}{9}

Apply the fraction rule:

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

= - 10 0^{\circ}

= 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

- θ = 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

- θ = 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

- θ = 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

Move

3 θ

to the left side

- θ = 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

Subtract

3 θ

from both sides

- θ - 3 θ = 3 θ + 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ} - 3 θ

Simplify

- 4 θ = 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

- 4 θ = 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

Divide both sides by

- 4

- 4 θ = 18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ}

Divide both sides by

- 4

\frac{- 4 θ}{- 4} = \frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{10 0 ^{\circ}}{- 4}

Simplify

\frac{- 4 θ}{- 4} = \frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{10 0 ^{\circ}}{- 4}

Simplify

\frac{- 4 θ}{- 4} : θ

\frac{- 4 θ}{- 4}

Apply the fraction rule:

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

= \frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{10 0 ^{\circ}}{- 4} : - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

\frac{18 0 ^{\circ}}{- 4} + \frac{36 0 ^{\circ} n}{- 4} - \frac{10 0 ^{\circ}}{- 4}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 10 0 ^{\circ}}{- 4}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 10 0 ^{\circ}}{4}

Join

18 0^{\circ} + 36 0^{\circ} n - 10 0^{\circ} : \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9}

18 0^{\circ} + 36 0^{\circ} n - 10 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} - 10 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 - 90 0 ^{\circ}}{9}

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

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

Add similar elements:

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

= 72 0^{\circ} + 2 \cdot 162 0^{\circ} n

Multiply the numbers:

2 \cdot 9 = 18

= 72 0^{\circ} + 324 0^{\circ} n

= \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9}

= - \frac{\frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9}}{4}

Simplify

\frac{\frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9}}{4} : \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{36}

\frac{\frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9}}{4}

Apply the fraction rule:

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

= \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{9 \cdot 4}

Multiply the numbers:

9 \cdot 4 = 36

= \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{36}

= - \frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{36}

Cancel

\frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{36} : \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

\frac{72 0 ^{\circ} + 324 0 ^{\circ} n}{36}

Factor

72 0^{\circ} + 324 0^{\circ} n : 36 0^{\circ} (2 + 9 n)

72 0^{\circ} + 324 0^{\circ} n

Rewrite as

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

Factor out common term

36 0^{\circ}

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

= \frac{36 0 ^{\circ} ( 2 + 9 n )}{36}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

= - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}, θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

θ = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{6}, θ = - \frac{18 0 ^{\circ} ( 9 n + 2 )}{18}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(20-θ)-cos(10+3θ)=0 ?
The general solution for sin(20-θ)-cos(10+3θ)=0 is θ=(1080n+180)/6 ,θ=-(180(9n+2))/(18)