What is the general solution for sin(x)=cos(19) ?

The general solution for sin(x)=cos(19) is x=360n+71,x=180-71+360n

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sin(x)=cos(19)

Solution

sin (x) = cos (1 9^{\circ})

Solution

x = 36 0^{\circ} n + 7 1^{\circ}, x = 18 0^{\circ} - 7 1^{\circ} + 36 0^{\circ} n

Radians

x = \frac{71 π}{180} + 2 πn, x = π - \frac{71 π}{180} + 2 πn

Solution steps

sin (x) = cos (1 9^{\circ})

Rewrite using trig identities

cos (1 9^{\circ})

Use the following identity:

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

sin (9 0^{\circ} - 1 9^{\circ})

sin (x) = sin (9 0^{\circ} - 1 9^{\circ})

Apply trig inverse properties

sin (x) = sin (9 0^{\circ} - 1 9^{\circ})

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

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

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

x = 9 0^{\circ} - 1 9^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 7 1^{\circ}

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

Simplify

9 0^{\circ} - 1 9^{\circ} + 36 0^{\circ} n : 36 0^{\circ} n + 7 1^{\circ}

9 0^{\circ} - 1 9^{\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} - 1 9^{\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 - 342 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 342 0^{\circ} = 1278 0^{\circ}

= 36 0^{\circ} n + 7 1^{\circ}

x = 36 0^{\circ} n + 7 1^{\circ}

x = 18 0^{\circ} - (9 0^{\circ} - 1 9^{\circ}) + 36 0^{\circ} n : x = 18 0^{\circ} - 7 1^{\circ} + 36 0^{\circ} n

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

Join

9 0^{\circ} - 1 9^{\circ} : 7 1^{\circ}

9 0^{\circ} - 1 9^{\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} - 1 9^{\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 - 342 0 ^{\circ}}{180}

Add similar elements:

1620 0^{\circ} - 342 0^{\circ} = 1278 0^{\circ}

= 7 1^{\circ}

x = 18 0^{\circ} - 7 1^{\circ} + 36 0^{\circ} n

x = 36 0^{\circ} n + 7 1^{\circ}, x = 18 0^{\circ} - 7 1^{\circ} + 36 0^{\circ} n

Popular Examples

1/(1.5)=(sin(30))/(sin(x)),x<90

\frac{1}{1.5} = \frac{sin ( 3 0 ^{\circ} )}{sin ( x )}, x < 9 0^{\circ}

tan(A)= 8/15

tan (A) = \frac{8}{15}

tan^2(θ)=sec^2(θ)+1

tan^{2} (θ) = sec^{2} (θ) + 1

cos(2B)= 21/35 ,cos(B)

cos (2 B) = \frac{21}{35}, cos (B)

2cos^4(x)+3sin^2(x)-2=0

2 cos^{4} (x) + 3 sin^{2} (x) - 2 = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=cos(19) ?
The general solution for sin(x)=cos(19) is x=360n+71,x=180-71+360n