What is the general solution for cos(θ/3+24)=sin(θ) ?

The general solution for cos(θ/3+24)=sin(θ) is θ=(1980+10800n)/(40),θ=(3420+10800n)/(20)

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cos(θ/3+24)=sin(θ)

Solution

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (θ)

Solution

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}, θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

Radians

θ = \frac{\frac{11 π}{5}}{8} + \frac{60 π}{40} n, θ = \frac{\frac{19 π}{5}}{4} + \frac{60 π}{20} n

Solution steps

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (θ)

Rewrite using trig identities

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (θ)

Use the following identity:

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

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}))

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}))

Apply trig inverse properties

cos (\frac{θ}{3} + 2 4^{\circ}) = sin (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}))

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

θ = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n : θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

θ = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n

Move

(\frac{θ}{3} + 2 4^{\circ})

to the left side

θ = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n

Add

(\frac{θ}{3} + 2 4^{\circ})

to both sides

θ + \frac{θ}{3} + 2 4^{\circ} = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n + \frac{θ}{3} + 2 4^{\circ}

Simplify

θ + \frac{θ}{3} + 2 4^{\circ} = 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n + \frac{θ}{3} + 2 4^{\circ}

Simplify

θ + \frac{θ}{3} + 2 4^{\circ} : \frac{20 θ + 36 0 ^{\circ}}{15}

θ + \frac{θ}{3} + 2 4^{\circ}

Convert element to fraction:

θ = \frac{θ}{1}

= \frac{θ}{3} + 2 4^{\circ} + \frac{θ}{1}

Least Common Multiplier of

3, 15, 1 : 15

3, 15, 1

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

3, 15, 1

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

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

15

For

\frac{θ}{3} :

multiply the denominator and numerator by

5

\frac{θ}{3} = \frac{θ \cdot 5}{3 \cdot 5} = \frac{θ \cdot 5}{15}

For

\frac{θ}{1} :

multiply the denominator and numerator by

15

\frac{θ}{1} = \frac{θ \cdot 15}{1 \cdot 15} = \frac{θ \cdot 15}{15}

= \frac{θ \cdot 5}{15} + 2 4^{\circ} + \frac{θ \cdot 15}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{θ \cdot 5 + 36 0 ^{\circ} + θ \cdot 15}{15}

θ \cdot 5 + 36 0^{\circ} + θ \cdot 15 = 20 θ + 36 0^{\circ}

θ \cdot 5 + 36 0^{\circ} + θ \cdot 15

Group like terms

= 5 θ + 15 θ + 36 0^{\circ}

Add similar elements:

5 θ + 15 θ = 20 θ

= 20 θ + 36 0^{\circ}

= \frac{20 θ + 36 0 ^{\circ}}{15}

Simplify

9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n + \frac{θ}{3} + 2 4^{\circ} : 9 0^{\circ} + 36 0^{\circ} n

9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) + 36 0^{\circ} n + \frac{θ}{3} + 2 4^{\circ}

Add similar elements:

- (\frac{θ}{3} + 2 4^{\circ}) + \frac{θ}{3} + 2 4^{\circ} = 0

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

\frac{20 θ + 36 0 ^{\circ}}{15} = 9 0^{\circ} + 36 0^{\circ} n

\frac{20 θ + 36 0 ^{\circ}}{15} = 9 0^{\circ} + 36 0^{\circ} n

\frac{20 θ + 36 0 ^{\circ}}{15} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

15

\frac{20 θ + 36 0 ^{\circ}}{15} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

15

\frac{15 ( 20 θ + 36 0 ^{\circ} )}{15} = 15 \cdot 9 0^{\circ} + 15 \cdot 36 0^{\circ} n

Simplify

\frac{15 ( 20 θ + 36 0 ^{\circ} )}{15} = 15 \cdot 9 0^{\circ} + 15 \cdot 36 0^{\circ} n

Simplify

\frac{15 ( 20 θ + 36 0 ^{\circ} )}{15} : 20 θ + 36 0^{\circ}

\frac{15 ( 20 θ + 36 0 ^{\circ} )}{15}

Divide the numbers:

\frac{15}{15} = 1

= 20 θ + 36 0^{\circ}

Simplify

15 \cdot 9 0^{\circ} + 15 \cdot 36 0^{\circ} n : 135 0^{\circ} + 540 0^{\circ} n

15 \cdot 9 0^{\circ} + 15 \cdot 36 0^{\circ} n

Multiply

15 \cdot 9 0^{\circ} : 135 0^{\circ}

15 \cdot 9 0^{\circ}

Multiply fractions:

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

= 135 0^{\circ}

= 135 0^{\circ} + 15 \cdot 36 0^{\circ} n

Multiply the numbers:

15 \cdot 2 = 30

= 135 0^{\circ} + 540 0^{\circ} n

20 θ + 36 0^{\circ} = 135 0^{\circ} + 540 0^{\circ} n

20 θ + 36 0^{\circ} = 135 0^{\circ} + 540 0^{\circ} n

20 θ + 36 0^{\circ} = 135 0^{\circ} + 540 0^{\circ} n

Move

36 0^{\circ}

to the right side

20 θ + 36 0^{\circ} = 135 0^{\circ} + 540 0^{\circ} n

Subtract

36 0^{\circ}

from both sides

20 θ + 36 0^{\circ} - 36 0^{\circ} = 135 0^{\circ} + 540 0^{\circ} n - 36 0^{\circ}

Simplify

20 θ = 135 0^{\circ} + 540 0^{\circ} n - 36 0^{\circ}

20 θ = 135 0^{\circ} + 540 0^{\circ} n - 36 0^{\circ}

Divide both sides by

20

20 θ = 135 0^{\circ} + 540 0^{\circ} n - 36 0^{\circ}

Divide both sides by

20

\frac{20 θ}{20} = \frac{135 0 ^{\circ}}{20} + \frac{540 0 ^{\circ} n}{20} - 1 8^{\circ}

Simplify

\frac{20 θ}{20} = \frac{135 0 ^{\circ}}{20} + \frac{540 0 ^{\circ} n}{20} - 1 8^{\circ}

Simplify

\frac{20 θ}{20} : θ

\frac{20 θ}{20}

Divide the numbers:

\frac{20}{20} = 1

= θ

Simplify

\frac{135 0 ^{\circ}}{20} + \frac{540 0 ^{\circ} n}{20} - 1 8^{\circ} : \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

\frac{135 0 ^{\circ}}{20} + \frac{540 0 ^{\circ} n}{20} - 1 8^{\circ}

Group like terms

= - 1 8^{\circ} + \frac{540 0 ^{\circ} n}{20} + \frac{135 0 ^{\circ}}{20}

Apply rule

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

= \frac{- 36 0 ^{\circ} + 540 0 ^{\circ} n + 135 0 ^{\circ}}{20}

Join

- 36 0^{\circ} + 540 0^{\circ} n + 135 0^{\circ} : \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{2}

- 36 0^{\circ} + 540 0^{\circ} n + 135 0^{\circ}

Convert element to fraction:

36 0^{\circ} = 36 0^{\circ}, 540 0^{\circ} n = \frac{540 0 ^{\circ} n 2}{2}

= - 36 0^{\circ} + \frac{540 0 ^{\circ} n \cdot 2}{2} + 135 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} 2 + 540 0 ^{\circ} n \cdot 2 + 270 0 ^{\circ}}{2}

- 36 0^{\circ} 2 + 540 0^{\circ} n \cdot 2 + 270 0^{\circ} = 198 0^{\circ} + 1080 0^{\circ} n

- 36 0^{\circ} 2 + 540 0^{\circ} n \cdot 2 + 270 0^{\circ}

Multiply the numbers:

2 \cdot 2 = 4

= - 72 0^{\circ} + 30 \cdot 36 0^{\circ} n + 270 0^{\circ}

Multiply the numbers:

30 \cdot 2 = 60

= - 72 0^{\circ} + 1080 0^{\circ} n + 270 0^{\circ}

Group like terms

= - 72 0^{\circ} + 270 0^{\circ} + 1080 0^{\circ} n

Add similar elements:

- 72 0^{\circ} + 270 0^{\circ} = 198 0^{\circ}

= 198 0^{\circ} + 1080 0^{\circ} n

= \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{2}

= \frac{\frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{2}}{20}

Apply the fraction rule:

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

= \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{2 \cdot 20}

Multiply the numbers:

2 \cdot 20 = 40

= \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}

θ = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n : θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n

Move

(9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}))

to the left side

θ = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n

Add

(9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}))

to both sides

θ + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})

Simplify

θ + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) = 18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})

Simplify

θ + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) : \frac{20 θ + 198 0 ^{\circ}}{30}

θ + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})

- (\frac{θ}{3} + 2 4^{\circ}) : - \frac{θ}{3} - 2 4^{\circ}

- (\frac{θ}{3} + 2 4^{\circ})

Distribute parentheses

= - (\frac{θ}{3}) - (2 4^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - \frac{θ}{3} - 2 4^{\circ}

= θ + 9 0^{\circ} - \frac{θ}{3} - 2 4^{\circ}

Simplify

θ + 9 0^{\circ} - \frac{θ}{3} - 2 4^{\circ} : \frac{20 θ + 198 0 ^{\circ}}{30}

θ + 9 0^{\circ} - \frac{θ}{3} - 2 4^{\circ}

Group like terms

= θ - \frac{θ}{3} + 9 0^{\circ} - 2 4^{\circ}

Convert element to fraction:

θ = \frac{θ 15}{15}

= - \frac{θ}{3} + 9 0^{\circ} - 2 4^{\circ} + \frac{θ \cdot 15}{15}

Least Common Multiplier of

3, 2, 15, 15 : 30

3, 2, 15, 15

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

Compute a number comprised of factors that appear in at least one of the following:

3, 2, 15, 15

= 3 \cdot 2 \cdot 5

Multiply the numbers:

3 \cdot 2 \cdot 5 = 30

= 30

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

30

For

\frac{θ}{3} :

multiply the denominator and numerator by

10

\frac{θ}{3} = \frac{θ \cdot 10}{3 \cdot 10} = \frac{θ \cdot 10}{30}

For

9 0^{\circ} :

multiply the denominator and numerator by

15

9 0^{\circ} = \frac{18 0 ^{\circ} 15}{2 \cdot 15} = 9 0^{\circ}

For

2 4^{\circ} :

multiply the denominator and numerator by

2

2 4^{\circ} = \frac{36 0 ^{\circ} 2}{15 \cdot 2} = 2 4^{\circ}

For

\frac{θ \cdot 15}{15} :

multiply the denominator and numerator by

2

\frac{θ \cdot 15}{15} = \frac{θ \cdot 15 \cdot 2}{15 \cdot 2} = \frac{30 θ}{30}

= - \frac{θ \cdot 10}{30} + 9 0^{\circ} - 2 4^{\circ} + \frac{30 θ}{30}

Since the denominators are equal, combine the fractions:

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

= \frac{- θ \cdot 10 + 18 0 ^{\circ} 15 - 72 0 ^{\circ} + 30 θ}{30}

- θ \cdot 10 + 18 0^{\circ} 15 - 72 0^{\circ} + 30 θ = 20 θ + 198 0^{\circ}

- θ \cdot 10 + 18 0^{\circ} 15 - 72 0^{\circ} + 30 θ

Group like terms

= - 10 θ + 30 θ + 270 0^{\circ} - 72 0^{\circ}

Add similar elements:

- 10 θ + 30 θ = 20 θ

= 20 θ + 270 0^{\circ} - 72 0^{\circ}

Add similar elements:

270 0^{\circ} - 72 0^{\circ} = 198 0^{\circ}

= 20 θ + 198 0^{\circ}

= \frac{20 θ + 198 0 ^{\circ}}{30}

= \frac{20 θ + 198 0 ^{\circ}}{30}

Simplify

18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) : 18 0^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 36 0^{\circ} n + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})

Add similar elements:

- (9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ})) + 9 0^{\circ} - (\frac{θ}{3} + 2 4^{\circ}) = 0

= 18 0^{\circ} + 36 0^{\circ} n

\frac{20 θ + 198 0 ^{\circ}}{30} = 18 0^{\circ} + 36 0^{\circ} n

\frac{20 θ + 198 0 ^{\circ}}{30} = 18 0^{\circ} + 36 0^{\circ} n

\frac{20 θ + 198 0 ^{\circ}}{30} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

30

\frac{20 θ + 198 0 ^{\circ}}{30} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

30

\frac{30 ( 20 θ + 198 0 ^{\circ} )}{30} = 540 0^{\circ} + 30 \cdot 36 0^{\circ} n

Simplify

20 θ + 198 0^{\circ} = 540 0^{\circ} + 1080 0^{\circ} n

20 θ + 198 0^{\circ} = 540 0^{\circ} + 1080 0^{\circ} n

Move

198 0^{\circ}

to the right side

20 θ + 198 0^{\circ} = 540 0^{\circ} + 1080 0^{\circ} n

Subtract

198 0^{\circ}

from both sides

20 θ + 198 0^{\circ} - 198 0^{\circ} = 540 0^{\circ} + 1080 0^{\circ} n - 198 0^{\circ}

Simplify

20 θ = 342 0^{\circ} + 1080 0^{\circ} n

20 θ = 342 0^{\circ} + 1080 0^{\circ} n

Divide both sides by

20

20 θ = 342 0^{\circ} + 1080 0^{\circ} n

Divide both sides by

20

\frac{20 θ}{20} = 17 1^{\circ} + \frac{1080 0 ^{\circ} n}{20}

Simplify

\frac{20 θ}{20} = 17 1^{\circ} + \frac{1080 0 ^{\circ} n}{20}

Simplify

\frac{20 θ}{20} : θ

\frac{20 θ}{20}

Divide the numbers:

\frac{20}{20} = 1

= θ

Simplify

17 1^{\circ} + \frac{1080 0 ^{\circ} n}{20} : \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

17 1^{\circ} + \frac{1080 0 ^{\circ} n}{20}

Apply rule

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

= \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}, θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

θ = \frac{198 0 ^{\circ} + 1080 0 ^{\circ} n}{40}, θ = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n}{20}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ/3+24)=sin(θ) ?
The general solution for cos(θ/3+24)=sin(θ) is θ=(1980+10800n)/(40),θ=(3420+10800n)/(20)