What is the general solution for sin(θ)=cos(2θ+54) ?

The general solution for sin(θ)=cos(2θ+54) is θ=(1800n+180}{15},θ=-\frac{720+1800n)/5

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sin(θ)=cos(2θ+54)

Solution

sin (θ) = cos (2 θ + 5 4^{\circ})

Solution

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}, θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

Radians

θ = \frac{\frac{π}{5}}{3} + \frac{10 π}{15} n, θ = - \frac{4 π}{5} - \frac{10 π}{5} n

Solution steps

sin (θ) = cos (2 θ + 5 4^{\circ})

Rewrite using trig identities

sin (θ) = cos (2 θ + 5 4^{\circ})

Use the following identity:

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

sin (θ) = sin (9 0^{\circ} - (2 θ + 5 4^{\circ}))

sin (θ) = sin (9 0^{\circ} - (2 θ + 5 4^{\circ}))

Apply trig inverse properties

sin (θ) = sin (9 0^{\circ} - (2 θ + 5 4^{\circ}))

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

θ = 9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n : θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

θ = 9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n

Expand

9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n : - 2 θ + 36 0^{\circ} n + 3 6^{\circ}

9 0^{\circ} - (2 θ + 5 4^{\circ}) + 36 0^{\circ} n

- (2 θ + 5 4^{\circ}) : - 2 θ - 5 4^{\circ}

- (2 θ + 5 4^{\circ})

Distribute parentheses

= - (2 θ) - (5 4^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - 2 θ - 5 4^{\circ}

= 9 0^{\circ} - 2 θ - 5 4^{\circ} + 36 0^{\circ} n

Simplify

9 0^{\circ} - 2 θ - 5 4^{\circ} + 36 0^{\circ} n : - 2 θ + 36 0^{\circ} n + 3 6^{\circ}

9 0^{\circ} - 2 θ - 5 4^{\circ} + 36 0^{\circ} n

Group like terms

= - 2 θ + 36 0^{\circ} n + 9 0^{\circ} - 5 4^{\circ}

Least Common Multiplier of

2, 10 : 10

2, 10

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

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

2

10

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

9 0^{\circ} :

multiply the denominator and numerator by

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

= 9 0^{\circ} - 5 4^{\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} 5 - 54 0 ^{\circ}}{10}

Add similar elements:

90 0^{\circ} - 54 0^{\circ} = 36 0^{\circ}

= 3 6^{\circ}

Cancel the common factor:

2

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

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

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

Move

2 θ

to the left side

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

Add

2 θ

to both sides

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

Simplify

3 θ = 36 0^{\circ} n + 3 6^{\circ}

3 θ = 36 0^{\circ} n + 3 6^{\circ}

Divide both sides by

3

3 θ = 36 0^{\circ} n + 3 6^{\circ}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{3 6 ^{\circ}}{3} : \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

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

Apply rule

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

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

Join

36 0^{\circ} n + 3 6^{\circ} : \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{5}

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

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 5}{5}

= \frac{36 0 ^{\circ} n \cdot 5}{5} + 3 6^{\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 5 + 18 0 ^{\circ}}{5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{5}

= \frac{\frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{5}}{3}

Apply the fraction rule:

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

= \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{5 \cdot 3}

Multiply the numbers:

5 \cdot 3 = 15

= \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}

θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n : θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n

Expand

18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} + 2 θ - 3 6^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (2 θ + 5 4^{\circ})) + 36 0^{\circ} n

Expand

9 0^{\circ} - (2 θ + 5 4^{\circ}) : - 2 θ + 3 6^{\circ}

9 0^{\circ} - (2 θ + 5 4^{\circ})

- (2 θ + 5 4^{\circ}) : - 2 θ - 5 4^{\circ}

- (2 θ + 5 4^{\circ})

Distribute parentheses

= - (2 θ) - (5 4^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - 2 θ - 5 4^{\circ}

= 9 0^{\circ} - 2 θ - 5 4^{\circ}

Simplify

9 0^{\circ} - 2 θ - 5 4^{\circ} : - 2 θ + 3 6^{\circ}

9 0^{\circ} - 2 θ - 5 4^{\circ}

Group like terms

= - 2 θ + 9 0^{\circ} - 5 4^{\circ}

Least Common Multiplier of

2, 10 : 10

2, 10

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

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

2

10

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

9 0^{\circ} :

multiply the denominator and numerator by

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

= 9 0^{\circ} - 5 4^{\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} 5 - 54 0 ^{\circ}}{10}

Add similar elements:

90 0^{\circ} - 54 0^{\circ} = 36 0^{\circ}

= 3 6^{\circ}

Cancel the common factor:

2

= - 2 θ + 3 6^{\circ}

= - 2 θ + 3 6^{\circ}

= 18 0^{\circ} - (- 2 θ + 3 6^{\circ}) + 36 0^{\circ} n

- (- 2 θ + 3 6^{\circ}) : 2 θ - 3 6^{\circ}

- (- 2 θ + 3 6^{\circ})

Distribute parentheses

= - (- 2 θ) - (3 6^{\circ})

Apply minus-plus rules

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

= 2 θ - 3 6^{\circ}

= 18 0^{\circ} + 2 θ - 3 6^{\circ} + 36 0^{\circ} n

θ = 18 0^{\circ} + 2 θ - 3 6^{\circ} + 36 0^{\circ} n

Move

2 θ

to the left side

θ = 18 0^{\circ} + 2 θ - 3 6^{\circ} + 36 0^{\circ} n

Subtract

2 θ

from both sides

θ - 2 θ = 18 0^{\circ} + 2 θ - 3 6^{\circ} + 36 0^{\circ} n - 2 θ

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

\frac{- θ}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{3 6 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

Simplify

\frac{- θ}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{3 6 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

Simplify

\frac{- θ}{- 1} : θ

\frac{- θ}{- 1}

Apply the fraction rule:

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

= \frac{θ}{1}

Apply rule

\frac{a}{1} = a

= θ

Simplify

\frac{18 0 ^{\circ}}{- 1} - \frac{3 6 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} : - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

\frac{18 0 ^{\circ}}{- 1} - \frac{3 6 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

Apply rule

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

= \frac{18 0 ^{\circ} - 3 6 ^{\circ} + 36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

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

= - \frac{18 0 ^{\circ} - 3 6 ^{\circ} + 36 0 ^{\circ} n}{1}

Join

18 0^{\circ} - 3 6^{\circ} + 36 0^{\circ} n : \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

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

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 5}{5}

= 18 0^{\circ} - 3 6^{\circ} + \frac{36 0 ^{\circ} n \cdot 5}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 5 - 18 0 ^{\circ} + 36 0 ^{\circ} n \cdot 5}{5}

18 0^{\circ} 5 - 18 0^{\circ} + 36 0^{\circ} n \cdot 5 = 72 0^{\circ} + 180 0^{\circ} n

18 0^{\circ} 5 - 18 0^{\circ} + 36 0^{\circ} n \cdot 5

Add similar elements:

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

= 72 0^{\circ} + 2 \cdot 90 0^{\circ} n

Multiply the numbers:

2 \cdot 5 = 10

= 72 0^{\circ} + 180 0^{\circ} n

= \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

= - \frac{\frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}, θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

θ = \frac{180 0 ^{\circ} n + 18 0 ^{\circ}}{15}, θ = - \frac{72 0 ^{\circ} + 180 0 ^{\circ} n}{5}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)=cos(2θ+54) ?
The general solution for sin(θ)=cos(2θ+54) is θ=(1800n+180}{15},θ=-\frac{720+1800n)/5