What is the general solution for sin(2x+4)=cos(46) ?

The general solution for sin(2x+4)=cos(46) is x=(16200n+1980-180)/(90),x=(3060+8100n-90)/(45)

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sin(2x+4)=cos(46)

Solution

sin (2 x + 4) = cos (4 6^{\circ})

Solution

x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}, x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

Radians

x = \frac{\frac{11 π}{5}}{18} - 2 + \frac{90 π}{90} n, x = - 2 + \frac{\frac{17 π}{5}}{9} + \frac{45 π}{45} n

Solution steps

sin (2 x + 4) = cos (4 6^{\circ})

Rewrite using trig identities

cos (4 6^{\circ})

Use the following identity:

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

sin (9 0^{\circ} - 4 6^{\circ})

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

Apply trig inverse properties

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

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

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

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

2 x + 4 = 9 0^{\circ} - 4 6^{\circ} + 36 0^{\circ} n : x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

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

Move

4

to the right side

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

Subtract

4

from both sides

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

Simplify

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

Simplify

2 x + 4 - 4 : 2 x

2 x + 4 - 4

Add similar elements:

4 - 4 = 0

= 2 x

Simplify

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

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

Combine the fractions

9 0^{\circ} - 4 6^{\circ} : 4 4^{\circ}

9 0^{\circ} - 4 6^{\circ}

Least Common Multiplier of

2, 90 : 90

2, 90

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

90 : 2 \cdot 3 \cdot 3 \cdot 5

90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3 \cdot 5

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

2

90

= 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}

= 9 0^{\circ} - 4 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} 45 - 414 0 ^{\circ}}{90}

Add similar elements:

810 0^{\circ} - 414 0^{\circ} = 396 0^{\circ}

= 4 4^{\circ}

Cancel the common factor:

2

= 4 4^{\circ}

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

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

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{2} + \frac{4 4 ^{\circ}}{2} - \frac{4}{2} : \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

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

Apply rule

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

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

Join

36 0^{\circ} n + 4 4^{\circ} - 4 : \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{45}

36 0^{\circ} n + 4 4^{\circ} - 4

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 45}{45}, 4 = \frac{4 \cdot 45}{45}

= \frac{36 0 ^{\circ} n \cdot 45}{45} + 4 4^{\circ} - \frac{4 \cdot 45}{45}

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 45 + 198 0 ^{\circ} - 4 \cdot 45}{45}

36 0^{\circ} n \cdot 45 + 198 0^{\circ} - 4 \cdot 45 = 1620 0^{\circ} n + 198 0^{\circ} - 180

36 0^{\circ} n \cdot 45 + 198 0^{\circ} - 4 \cdot 45

Multiply the numbers:

2 \cdot 45 = 90

= 1620 0^{\circ} n + 198 0^{\circ} - 4 \cdot 45

Multiply the numbers:

4 \cdot 45 = 180

= 1620 0^{\circ} n + 198 0^{\circ} - 180

= \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{45}

= \frac{\frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{45}}{2}

Apply the fraction rule:

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

= \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{45 \cdot 2}

Multiply the numbers:

45 \cdot 2 = 90

= \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}

2 x + 4 = 18 0^{\circ} - (9 0^{\circ} - 4 6^{\circ}) + 36 0^{\circ} n : x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

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

Move

4

to the right side

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

Subtract

4

from both sides

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

Simplify

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

Simplify

2 x + 4 - 4 : 2 x

2 x + 4 - 4

Add similar elements:

4 - 4 = 0

= 2 x

Simplify

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

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

Join

9 0^{\circ} - 4 6^{\circ} : 4 4^{\circ}

9 0^{\circ} - 4 6^{\circ}

Least Common Multiplier of

2, 90 : 90

2, 90

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

90 : 2 \cdot 3 \cdot 3 \cdot 5

90

90

divides by

290 = 45 \cdot 2

= 2 \cdot 45

45

divides by

345 = 15 \cdot 3

= 2 \cdot 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3 \cdot 5

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

2

90

= 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}

= 9 0^{\circ} - 4 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} 45 - 414 0 ^{\circ}}{90}

Add similar elements:

810 0^{\circ} - 414 0^{\circ} = 396 0^{\circ}

= 4 4^{\circ}

Cancel the common factor:

2

= 4 4^{\circ}

= 18 0^{\circ} - 4 4^{\circ} + 36 0^{\circ} n - 4

2 x = 18 0^{\circ} - 4 4^{\circ} + 36 0^{\circ} n - 4

2 x = 18 0^{\circ} - 4 4^{\circ} + 36 0^{\circ} n - 4

2 x = 18 0^{\circ} - 4 4^{\circ} + 36 0^{\circ} n - 4

Divide both sides by

2

2 x = 18 0^{\circ} - 4 4^{\circ} + 36 0^{\circ} n - 4

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} - \frac{4 4 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{4}{2} : \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

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

Group like terms

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

Apply rule

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

= \frac{18 0 ^{\circ} - 4 + 36 0 ^{\circ} n - 4 4 ^{\circ}}{2}

Join

18 0^{\circ} - 4 + 36 0^{\circ} n - 4 4^{\circ} : \frac{612 0 ^{\circ} + 1620 0 ^{\circ} n - 180}{45}

18 0^{\circ} - 4 + 36 0^{\circ} n - 4 4^{\circ}

Convert element to fraction:

18 0^{\circ} = 18 0^{\circ}, 4 = \frac{4 \cdot 45}{45}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 45}{45}

= 18 0^{\circ} - \frac{4 \cdot 45}{45} + \frac{36 0 ^{\circ} n \cdot 45}{45} - 4 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} 45 - 4 \cdot 45 + 36 0 ^{\circ} n \cdot 45 - 198 0 ^{\circ}}{45}

18 0^{\circ} 45 - 4 \cdot 45 + 36 0^{\circ} n \cdot 45 - 198 0^{\circ} = 612 0^{\circ} + 1620 0^{\circ} n - 180

18 0^{\circ} 45 - 4 \cdot 45 + 36 0^{\circ} n \cdot 45 - 198 0^{\circ}

Group like terms

= 810 0^{\circ} - 198 0^{\circ} + 2 \cdot 810 0^{\circ} n - 4 \cdot 45

Add similar elements:

810 0^{\circ} - 198 0^{\circ} = 612 0^{\circ}

= 612 0^{\circ} + 2 \cdot 810 0^{\circ} n - 4 \cdot 45

Multiply the numbers:

2 \cdot 45 = 90

= 612 0^{\circ} + 1620 0^{\circ} n - 4 \cdot 45

Multiply the numbers:

4 \cdot 45 = 180

= 612 0^{\circ} + 1620 0^{\circ} n - 180

= \frac{612 0 ^{\circ} + 1620 0 ^{\circ} n - 180}{45}

= \frac{\frac{612 0 ^{\circ} + 1620 0 ^{\circ} n - 180}{45}}{2}

Apply the fraction rule:

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

= \frac{612 0 ^{\circ} + 1620 0 ^{\circ} n - 180}{45 \cdot 2}

Multiply the numbers:

45 \cdot 2 = 90

= \frac{612 0 ^{\circ} + 1620 0 ^{\circ} n - 180}{90}

Factor

612 0^{\circ} + 1620 0^{\circ} n - 180 : 2 (306 0^{\circ} + 810 0^{\circ} n - 90)

612 0^{\circ} + 1620 0^{\circ} n - 180

Rewrite as

= 2 \cdot 306 0^{\circ} + 2 \cdot 810 0^{\circ} n - 2 \cdot 90

Factor out common term

2

= 2 (306 0^{\circ} + 810 0^{\circ} n - 90)

= \frac{2 ( 306 0 ^{\circ} + 810 0 ^{\circ} n - 90 )}{90}

Cancel the common factor:

2

= \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

x = \frac{1620 0 ^{\circ} n + 198 0 ^{\circ} - 180}{90}, x = \frac{306 0 ^{\circ} + 810 0 ^{\circ} n - 90}{45}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x+4)=cos(46) ?
The general solution for sin(2x+4)=cos(46) is x=(16200n+1980-180)/(90),x=(3060+8100n-90)/(45)