What is the general solution for sin(x)=cos(32) ?

The general solution for sin(x)=cos(32) is x=360n+58,x=180-58+360n

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sin(x)=cos(32)

Solution

sin (x) = cos (3 2^{\circ})

Solution

x = 36 0^{\circ} n + 5 8^{\circ}, x = 18 0^{\circ} - 5 8^{\circ} + 36 0^{\circ} n

Radians

x = \frac{29 π}{90} + 2 πn, x = π - \frac{29 π}{90} + 2 πn

Solution steps

sin (x) = cos (3 2^{\circ})

Rewrite using trig identities

cos (3 2^{\circ})

Use the following identity:

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

sin (9 0^{\circ} - 3 2^{\circ})

sin (x) = sin (9 0^{\circ} - 3 2^{\circ})

Apply trig inverse properties

sin (x) = sin (9 0^{\circ} - 3 2^{\circ})

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

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

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

x = 9 0^{\circ} - 3 2^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 5 8^{\circ}

x = 9 0^{\circ} - 3 2^{\circ} + 36 0^{\circ} n

Simplify

9 0^{\circ} - 3 2^{\circ} + 36 0^{\circ} n : 36 0^{\circ} n + 5 8^{\circ}

9 0^{\circ} - 3 2^{\circ} + 36 0^{\circ} n

Least Common Multiplier of

2, 45 : 90

2, 45

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

45 : 3 \cdot 3 \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

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

2

45

= 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 3 \cdot 5 = 90

= 90

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

90

For

9 0^{\circ} :

multiply the denominator and numerator by

45

9 0^{\circ} = \frac{18 0 ^{\circ} 45}{2 \cdot 45} = 9 0^{\circ}

For

3 2^{\circ} :

multiply the denominator and numerator by

2

3 2^{\circ} = \frac{144 0 ^{\circ} 2}{45 \cdot 2} = 3 2^{\circ}

= 9 0^{\circ} - 3 2^{\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} 45 - 288 0 ^{\circ}}{90}

Add similar elements:

810 0^{\circ} - 288 0^{\circ} = 522 0^{\circ}

= 36 0^{\circ} n + 5 8^{\circ}

x = 36 0^{\circ} n + 5 8^{\circ}

x = 18 0^{\circ} - (9 0^{\circ} - 3 2^{\circ}) + 36 0^{\circ} n : x = 18 0^{\circ} - 5 8^{\circ} + 36 0^{\circ} n

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

Join

9 0^{\circ} - 3 2^{\circ} : 5 8^{\circ}

9 0^{\circ} - 3 2^{\circ}

Least Common Multiplier of

2, 45 : 90

2, 45

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

45 : 3 \cdot 3 \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

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

2

45

= 2 \cdot 3 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \cdot 3 \cdot 5 = 90

= 90

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

90

For

9 0^{\circ} :

multiply the denominator and numerator by

45

9 0^{\circ} = \frac{18 0 ^{\circ} 45}{2 \cdot 45} = 9 0^{\circ}

For

3 2^{\circ} :

multiply the denominator and numerator by

2

3 2^{\circ} = \frac{144 0 ^{\circ} 2}{45 \cdot 2} = 3 2^{\circ}

= 9 0^{\circ} - 3 2^{\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} 45 - 288 0 ^{\circ}}{90}

Add similar elements:

810 0^{\circ} - 288 0^{\circ} = 522 0^{\circ}

= 5 8^{\circ}

x = 18 0^{\circ} - 5 8^{\circ} + 36 0^{\circ} n

x = 36 0^{\circ} n + 5 8^{\circ}, x = 18 0^{\circ} - 5 8^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=cos(32) ?
The general solution for sin(x)=cos(32) is x=360n+58,x=180-58+360n