What is the general solution for sin(θ)=cos(37) ?

The general solution for sin(θ)=cos(37) is θ=360n+53,θ=180-53+360n

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sin(θ)=cos(37)

Solution

sin (θ) = cos (3 7^{\circ})

Solution

θ = 36 0^{\circ} n + 5 3^{\circ}, θ = 18 0^{\circ} - 5 3^{\circ} + 36 0^{\circ} n

Radians

θ = \frac{53 π}{180} + 2 πn, θ = π - \frac{53 π}{180} + 2 πn

Solution steps

sin (θ) = cos (3 7^{\circ})

Rewrite using trig identities

cos (3 7^{\circ})

Use the following identity:

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

sin (9 0^{\circ} - 3 7^{\circ})

sin (θ) = sin (9 0^{\circ} - 3 7^{\circ})

Apply trig inverse properties

sin (θ) = sin (9 0^{\circ} - 3 7^{\circ})

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

θ = 9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - 3 7^{\circ}) + 36 0^{\circ} n

θ = 9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - 3 7^{\circ}) + 36 0^{\circ} n

θ = 9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n : θ = 36 0^{\circ} n + 5 3^{\circ}

θ = 9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n

Simplify

9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n : 36 0^{\circ} n + 5 3^{\circ}

9 0^{\circ} - 3 7^{\circ} + 36 0^{\circ} n

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

180 : 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

180

180

divides by

2180 = 90 \cdot 2

= 2 \cdot 90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

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

2

180

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

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

180

For

9 0^{\circ} :

multiply the denominator and numerator by

90

9 0^{\circ} = \frac{18 0 ^{\circ} 90}{2 \cdot 90} = 9 0^{\circ}

= 9 0^{\circ} - 3 7^{\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} 90 - 666 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 666 0^{\circ} = 954 0^{\circ}

= 36 0^{\circ} n + 5 3^{\circ}

θ = 36 0^{\circ} n + 5 3^{\circ}

θ = 18 0^{\circ} - (9 0^{\circ} - 3 7^{\circ}) + 36 0^{\circ} n : θ = 18 0^{\circ} - 5 3^{\circ} + 36 0^{\circ} n

θ = 18 0^{\circ} - (9 0^{\circ} - 3 7^{\circ}) + 36 0^{\circ} n

Join

9 0^{\circ} - 3 7^{\circ} : 5 3^{\circ}

9 0^{\circ} - 3 7^{\circ}

Least Common Multiplier of

2, 180 : 180

2, 180

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

180 : 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

180

180

divides by

2180 = 90 \cdot 2

= 2 \cdot 90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

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

2

180

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 180

= 180

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

180

For

9 0^{\circ} :

multiply the denominator and numerator by

90

9 0^{\circ} = \frac{18 0 ^{\circ} 90}{2 \cdot 90} = 9 0^{\circ}

= 9 0^{\circ} - 3 7^{\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} 90 - 666 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 666 0^{\circ} = 954 0^{\circ}

= 5 3^{\circ}

θ = 18 0^{\circ} - 5 3^{\circ} + 36 0^{\circ} n

θ = 36 0^{\circ} n + 5 3^{\circ}, θ = 18 0^{\circ} - 5 3^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)=cos(37) ?
The general solution for sin(θ)=cos(37) is θ=360n+53,θ=180-53+360n