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

The general solution for cos(θ)=sin(θ/3-10) is θ=(3240n+900}{12},θ=-\frac{900+3240n)/6

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Popular Trigonometry >

cos(θ)=sin(θ/3-10)

Solution

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

Solution

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}, θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

Radians

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

Solution steps

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

Rewrite using trig identities

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

Use the following identity:

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

cos (θ) = sin (9 0^{\circ} - θ)

cos (θ) = sin (9 0^{\circ} - θ)

Apply trig inverse properties

cos (θ) = sin (9 0^{\circ} - θ)

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

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

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

\frac{θ}{3} - 1 0^{\circ} = 9 0^{\circ} - θ + 36 0^{\circ} n : θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

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

Move

1 0^{\circ}

to the right side

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

Add

1 0^{\circ}

to both sides

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

Simplify

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

Simplify

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

\frac{θ}{3} - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

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

= \frac{θ}{3}

Simplify

9 0^{\circ} - θ + 36 0^{\circ} n + 1 0^{\circ} : - θ + 36 0^{\circ} n + 10 0^{\circ}

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

Group like terms

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

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} = 180 0^{\circ}

= 10 0^{\circ}

Cancel the common factor:

2

= - θ + 36 0^{\circ} n + 10 0^{\circ}

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

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

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{θ}{3} \cdot 3 : θ

\frac{θ}{3} \cdot 3

Multiply fractions:

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

= \frac{θ \cdot 3}{3}

Cancel the common factor:

3

= θ

Simplify

θ \cdot 3 : 3 θ

θ \cdot 3

Apply the commutative law:

θ \cdot 3 = 3 θ

3 θ

Simplify

36 0^{\circ} n \cdot 3 : 108 0^{\circ} n

36 0^{\circ} n \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 108 0^{\circ} n

Simplify

10 0^{\circ} \cdot 3 : 30 0^{\circ}

10 0^{\circ} \cdot 3

Multiply fractions:

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

= 30 0^{\circ}

Multiply the numbers:

5 \cdot 3 = 15

= 30 0^{\circ}

Cancel the common factor:

3

= 30 0^{\circ}

θ = - 3 θ + 108 0^{\circ} n + 30 0^{\circ}

θ = - 3 θ + 108 0^{\circ} n + 30 0^{\circ}

θ = - 3 θ + 108 0^{\circ} n + 30 0^{\circ}

Move

3 θ

to the left side

θ = - 3 θ + 108 0^{\circ} n + 30 0^{\circ}

Add

3 θ

to both sides

θ + 3 θ = - 3 θ + 108 0^{\circ} n + 30 0^{\circ} + 3 θ

Simplify

4 θ = 108 0^{\circ} n + 30 0^{\circ}

4 θ = 108 0^{\circ} n + 30 0^{\circ}

Divide both sides by

4

4 θ = 108 0^{\circ} n + 30 0^{\circ}

Divide both sides by

4

\frac{4 θ}{4} = \frac{108 0 ^{\circ} n}{4} + \frac{30 0 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} = \frac{108 0 ^{\circ} n}{4} + \frac{30 0 ^{\circ}}{4}

Simplify

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Divide the numbers:

\frac{4}{4} = 1

= θ

Simplify

\frac{108 0 ^{\circ} n}{4} + \frac{30 0 ^{\circ}}{4} : \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

\frac{108 0 ^{\circ} n}{4} + \frac{30 0 ^{\circ}}{4}

Apply rule

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

= \frac{108 0 ^{\circ} n + 30 0 ^{\circ}}{4}

Join

108 0^{\circ} n + 30 0^{\circ} : \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{3}

108 0^{\circ} n + 30 0^{\circ}

Convert element to fraction:

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

= \frac{108 0 ^{\circ} n \cdot 3}{3} + 30 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{108 0 ^{\circ} n \cdot 3 + 90 0 ^{\circ}}{3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{3}

= \frac{\frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{3}}{4}

Apply the fraction rule:

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

= \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}

\frac{θ}{3} - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - θ) + 36 0^{\circ} n : θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

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

Move

1 0^{\circ}

to the right side

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

Add

1 0^{\circ}

to both sides

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

Simplify

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

Simplify

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

\frac{θ}{3} - 1 0^{\circ} + 1 0^{\circ}

Add similar elements:

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

= \frac{θ}{3}

Simplify

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

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

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

- (9 0^{\circ} - θ)

Distribute parentheses

= - (9 0^{\circ}) - (- θ)

Apply minus-plus rules

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

= - 9 0^{\circ} + θ

= 18 0^{\circ} - 9 0^{\circ} + θ + 36 0^{\circ} n + 1 0^{\circ}

Simplify

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

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

Group like terms

= θ + 18 0^{\circ} + 36 0^{\circ} n - 9 0^{\circ} + 1 0^{\circ}

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}

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

Apply the fraction rule:

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

= - 8 0^{\circ}

Cancel the common factor:

2

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

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

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

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

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{θ}{3} \cdot 3 : θ

\frac{θ}{3} \cdot 3

Multiply fractions:

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

= \frac{θ \cdot 3}{3}

Cancel the common factor:

3

= θ

Simplify

θ \cdot 3 : 3 θ

θ \cdot 3

Apply the commutative law:

θ \cdot 3 = 3 θ

3 θ

Simplify

18 0^{\circ} 3 : 54 0^{\circ}

18 0^{\circ} 3

Apply the commutative law:

18 0^{\circ} 3 = 54 0^{\circ}

54 0^{\circ}

Simplify

36 0^{\circ} n \cdot 3 : 108 0^{\circ} n

36 0^{\circ} n \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 108 0^{\circ} n

Simplify

- 8 0^{\circ} \cdot 3 : - 24 0^{\circ}

- 8 0^{\circ} \cdot 3

Multiply fractions:

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

= - 24 0^{\circ}

Multiply the numbers:

4 \cdot 3 = 12

= - 24 0^{\circ}

Cancel the common factor:

3

= - 24 0^{\circ}

θ = 3 θ + 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

θ = 3 θ + 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

θ = 3 θ + 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

Move

3 θ

to the left side

θ = 3 θ + 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

Subtract

3 θ

from both sides

θ - 3 θ = 3 θ + 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ} - 3 θ

Simplify

- 2 θ = 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

- 2 θ = 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

Divide both sides by

- 2

- 2 θ = 54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

Divide both sides by

- 2

\frac{- 2 θ}{- 2} = \frac{54 0 ^{\circ}}{- 2} + \frac{108 0 ^{\circ} n}{- 2} - \frac{24 0 ^{\circ}}{- 2}

Simplify

\frac{- 2 θ}{- 2} = \frac{54 0 ^{\circ}}{- 2} + \frac{108 0 ^{\circ} n}{- 2} - \frac{24 0 ^{\circ}}{- 2}

Simplify

\frac{- 2 θ}{- 2} : θ

\frac{- 2 θ}{- 2}

Apply the fraction rule:

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

= \frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{54 0 ^{\circ}}{- 2} + \frac{108 0 ^{\circ} n}{- 2} - \frac{24 0 ^{\circ}}{- 2} : - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

\frac{54 0 ^{\circ}}{- 2} + \frac{108 0 ^{\circ} n}{- 2} - \frac{24 0 ^{\circ}}{- 2}

Apply rule

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

= \frac{54 0 ^{\circ} + 108 0 ^{\circ} n - 24 0 ^{\circ}}{- 2}

Apply the fraction rule:

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

= - \frac{54 0 ^{\circ} + 108 0 ^{\circ} n - 24 0 ^{\circ}}{2}

Join

54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ} : \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3}

54 0^{\circ} + 108 0^{\circ} n - 24 0^{\circ}

Convert element to fraction:

54 0^{\circ} = 54 0^{\circ}, 108 0^{\circ} n = \frac{108 0 ^{\circ} n 3}{3}

= 54 0^{\circ} + \frac{108 0 ^{\circ} n \cdot 3}{3} - 24 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{54 0 ^{\circ} 3 + 108 0 ^{\circ} n \cdot 3 - 72 0 ^{\circ}}{3}

54 0^{\circ} 3 + 108 0^{\circ} n \cdot 3 - 72 0^{\circ} = 90 0^{\circ} + 324 0^{\circ} n

54 0^{\circ} 3 + 108 0^{\circ} n \cdot 3 - 72 0^{\circ}

Multiply the numbers:

3 \cdot 3 = 9

= 162 0^{\circ} + 6 \cdot 54 0^{\circ} n - 72 0^{\circ}

Multiply the numbers:

6 \cdot 3 = 18

= 162 0^{\circ} + 324 0^{\circ} n - 72 0^{\circ}

Group like terms

= 162 0^{\circ} - 72 0^{\circ} + 324 0^{\circ} n

Add similar elements:

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

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

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3}

= - \frac{\frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3}}{2}

Simplify

\frac{\frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3}}{2} : \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

\frac{\frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3}}{2}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

= - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}, θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

θ = \frac{324 0 ^{\circ} n + 90 0 ^{\circ}}{12}, θ = - \frac{90 0 ^{\circ} + 324 0 ^{\circ} n}{6}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ)=sin(θ/3-10) ?
The general solution for cos(θ)=sin(θ/3-10) is θ=(3240n+900}{12},θ=-\frac{900+3240n)/6