What is the general solution for sin(3x+10)=cos(x+24) ?

The general solution for sin(3x+10)=cos(x+24) is x=(10800n+1980-300)/(120),x=(3420+10800n-300)/(60)

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sin(3x+10)=cos(x+24)

Solution

sin (3 x + 10) = cos (x + 2 4^{\circ})

Solution

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}, x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

Radians

x = \frac{\frac{11 π}{5}}{24} - \frac{5}{2} + \frac{60 π}{120} n, x = - 5 + \frac{\frac{19 π}{5}}{12} + \frac{60 π}{60} n

Solution steps

sin (3 x + 10) = cos (x + 2 4^{\circ})

Rewrite using trig identities

sin (3 x + 10) = cos (x + 2 4^{\circ})

Use the following identity:

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

sin (3 x + 10) = sin (9 0^{\circ} - (x + 2 4^{\circ}))

sin (3 x + 10) = sin (9 0^{\circ} - (x + 2 4^{\circ}))

Apply trig inverse properties

sin (3 x + 10) = sin (9 0^{\circ} - (x + 2 4^{\circ}))

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

3 x + 10 = 9 0^{\circ} - (x + 2 4^{\circ}) + 36 0^{\circ} n, 3 x + 10 = 18 0^{\circ} - (9 0^{\circ} - (x + 2 4^{\circ})) + 36 0^{\circ} n

3 x + 10 = 9 0^{\circ} - (x + 2 4^{\circ}) + 36 0^{\circ} n, 3 x + 10 = 18 0^{\circ} - (9 0^{\circ} - (x + 2 4^{\circ})) + 36 0^{\circ} n

3 x + 10 = 9 0^{\circ} - (x + 2 4^{\circ}) + 36 0^{\circ} n : x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

3 x + 10 = 9 0^{\circ} - (x + 2 4^{\circ}) + 36 0^{\circ} n

Expand

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

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

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

- (x + 2 4^{\circ})

Distribute parentheses

= - (x) - (2 4^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - x - 2 4^{\circ}

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

Simplify

9 0^{\circ} - x - 2 4^{\circ} + 36 0^{\circ} n : - x + 36 0^{\circ} n + 6 6^{\circ}

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

Group like terms

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

Least Common Multiplier of

2, 15 : 30

2, 15

Least Common Multiplier (LCM)

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

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

2

15

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \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

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}

= 9 0^{\circ} - 2 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} 15 - 72 0 ^{\circ}}{30}

Add similar elements:

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

= - x + 36 0^{\circ} n + 6 6^{\circ}

= - x + 36 0^{\circ} n + 6 6^{\circ}

3 x + 10 = - x + 36 0^{\circ} n + 6 6^{\circ}

Move

10

to the right side

3 x + 10 = - x + 36 0^{\circ} n + 6 6^{\circ}

Subtract

10

from both sides

3 x + 10 - 10 = - x + 36 0^{\circ} n + 6 6^{\circ} - 10

Simplify

3 x = - x + 36 0^{\circ} n + 6 6^{\circ} - 10

3 x = - x + 36 0^{\circ} n + 6 6^{\circ} - 10

Move

x

to the left side

3 x = - x + 36 0^{\circ} n + 6 6^{\circ} - 10

Add

x

to both sides

3 x + x = - x + 36 0^{\circ} n + 6 6^{\circ} - 10 + x

Simplify

4 x = 36 0^{\circ} n + 6 6^{\circ} - 10

4 x = 36 0^{\circ} n + 6 6^{\circ} - 10

Divide both sides by

4

4 x = 36 0^{\circ} n + 6 6^{\circ} - 10

Divide both sides by

4

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{6 6 ^{\circ}}{4} - \frac{10}{4}

Simplify

\frac{4 x}{4} = \frac{36 0 ^{\circ} n}{4} + \frac{6 6 ^{\circ}}{4} - \frac{10}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{4} + \frac{6 6 ^{\circ}}{4} - \frac{10}{4} : \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

\frac{36 0 ^{\circ} n}{4} + \frac{6 6 ^{\circ}}{4} - \frac{10}{4}

Apply rule

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

= \frac{36 0 ^{\circ} n + 6 6 ^{\circ} - 10}{4}

Join

36 0^{\circ} n + 6 6^{\circ} - 10 : \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{30}

36 0^{\circ} n + 6 6^{\circ} - 10

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 30}{30}, 10 = \frac{10 \cdot 30}{30}

= \frac{36 0 ^{\circ} n \cdot 30}{30} + 6 6^{\circ} - \frac{10 \cdot 30}{30}

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 30 + 198 0 ^{\circ} - 10 \cdot 30}{30}

36 0^{\circ} n \cdot 30 + 198 0^{\circ} - 10 \cdot 30 = 1080 0^{\circ} n + 198 0^{\circ} - 300

36 0^{\circ} n \cdot 30 + 198 0^{\circ} - 10 \cdot 30

Multiply the numbers:

2 \cdot 30 = 60

= 1080 0^{\circ} n + 198 0^{\circ} - 10 \cdot 30

Multiply the numbers:

10 \cdot 30 = 300

= 1080 0^{\circ} n + 198 0^{\circ} - 300

= \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{30}

= \frac{\frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{30}}{4}

Apply the fraction rule:

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

= \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{30 \cdot 4}

Multiply the numbers:

30 \cdot 4 = 120

= \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}

3 x + 10 = 18 0^{\circ} - (9 0^{\circ} - (x + 2 4^{\circ})) + 36 0^{\circ} n : x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

3 x + 10 = 18 0^{\circ} - (9 0^{\circ} - (x + 2 4^{\circ})) + 36 0^{\circ} n

Expand

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

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

Expand

9 0^{\circ} - (x + 2 4^{\circ}) : - x + 6 6^{\circ}

9 0^{\circ} - (x + 2 4^{\circ})

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

- (x + 2 4^{\circ})

Distribute parentheses

= - (x) - (2 4^{\circ})

Apply minus-plus rules

+ (- a) = - a

= - x - 2 4^{\circ}

= 9 0^{\circ} - x - 2 4^{\circ}

Simplify

9 0^{\circ} - x - 2 4^{\circ} : - x + 6 6^{\circ}

9 0^{\circ} - x - 2 4^{\circ}

Group like terms

= - x + 9 0^{\circ} - 2 4^{\circ}

Least Common Multiplier of

2, 15 : 30

2, 15

Least Common Multiplier (LCM)

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

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

2

15

= 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 3 \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

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}

= 9 0^{\circ} - 2 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} 15 - 72 0 ^{\circ}}{30}

Add similar elements:

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

= - x + 6 6^{\circ}

= - x + 6 6^{\circ}

= 18 0^{\circ} - (- x + 6 6^{\circ}) + 36 0^{\circ} n

- (- x + 6 6^{\circ}) : x - 6 6^{\circ}

- (- x + 6 6^{\circ})

Distribute parentheses

= - (- x) - (6 6^{\circ})

Apply minus-plus rules

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

= x - 6 6^{\circ}

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

3 x + 10 = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n

Move

10

to the right side

3 x + 10 = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n

Subtract

10

from both sides

3 x + 10 - 10 = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n - 10

Simplify

3 x = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n - 10

3 x = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n - 10

Move

x

to the left side

3 x = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n - 10

Subtract

x

from both sides

3 x - x = 18 0^{\circ} + x - 6 6^{\circ} + 36 0^{\circ} n - 10 - x

Simplify

2 x = 18 0^{\circ} - 6 6^{\circ} + 36 0^{\circ} n - 10

2 x = 18 0^{\circ} - 6 6^{\circ} + 36 0^{\circ} n - 10

Divide both sides by

2

2 x = 18 0^{\circ} - 6 6^{\circ} + 36 0^{\circ} n - 10

Divide both sides by

2

\frac{2 x}{2} = 9 0^{\circ} - \frac{6 6 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{10}{2}

Simplify

\frac{2 x}{2} = 9 0^{\circ} - \frac{6 6 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{10}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} - \frac{6 6 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{10}{2} : \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

9 0^{\circ} - \frac{6 6 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{10}{2}

Group like terms

= 9 0^{\circ} - \frac{10}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{6 6 ^{\circ}}{2}

Apply rule

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

= \frac{18 0 ^{\circ} - 10 + 36 0 ^{\circ} n - 6 6 ^{\circ}}{2}

Join

18 0^{\circ} - 10 + 36 0^{\circ} n - 6 6^{\circ} : \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{30}

18 0^{\circ} - 10 + 36 0^{\circ} n - 6 6^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 10 = \frac{10 \cdot 30}{30}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 30}{30}

= 18 0^{\circ} - \frac{10 \cdot 30}{30} + \frac{36 0 ^{\circ} n \cdot 30}{30} - 6 6^{\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} 30 - 10 \cdot 30 + 36 0 ^{\circ} n \cdot 30 - 198 0 ^{\circ}}{30}

18 0^{\circ} 30 - 10 \cdot 30 + 36 0^{\circ} n \cdot 30 - 198 0^{\circ} = 342 0^{\circ} + 1080 0^{\circ} n - 300

18 0^{\circ} 30 - 10 \cdot 30 + 36 0^{\circ} n \cdot 30 - 198 0^{\circ}

Group like terms

= 540 0^{\circ} - 198 0^{\circ} + 2 \cdot 540 0^{\circ} n - 10 \cdot 30

Add similar elements:

540 0^{\circ} - 198 0^{\circ} = 342 0^{\circ}

= 342 0^{\circ} + 2 \cdot 540 0^{\circ} n - 10 \cdot 30

Multiply the numbers:

2 \cdot 30 = 60

= 342 0^{\circ} + 1080 0^{\circ} n - 10 \cdot 30

Multiply the numbers:

10 \cdot 30 = 300

= 342 0^{\circ} + 1080 0^{\circ} n - 300

= \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{30}

= \frac{\frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{30}}{2}

Apply the fraction rule:

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

= \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{30 \cdot 2}

Multiply the numbers:

30 \cdot 2 = 60

= \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}, x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

x = \frac{1080 0 ^{\circ} n + 198 0 ^{\circ} - 300}{120}, x = \frac{342 0 ^{\circ} + 1080 0 ^{\circ} n - 300}{60}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x+10)=cos(x+24) ?
The general solution for sin(3x+10)=cos(x+24) is x=(10800n+1980-300)/(120),x=(3420+10800n-300)/(60)