What is the general solution for sin(2x)=cos(40) ?

The general solution for sin(2x)=cos(40) is x=(900+6480n)/(36),x=(2340+6480n)/(36)

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sin(2x)=cos(40)

Solution

sin (2 x) = cos (4 0^{\circ})

Solution

x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}, x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

Radians

x = \frac{5 π}{36} + \frac{36 π}{36} n, x = \frac{13 π}{36} + \frac{36 π}{36} n

Solution steps

sin (2 x) = cos (4 0^{\circ})

Rewrite using trig identities

cos (4 0^{\circ})

Use the following identity:

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

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

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

Apply trig inverse properties

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

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

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

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

2 x = 9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n : x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{9 0 ^{\circ}}{2} - \frac{4 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} : \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

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

Apply rule

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

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

Join

9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n : \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}

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

Convert element to fraction:

36 0^{\circ} n = \frac{36 0 ^{\circ} n}{1}

= 9 0^{\circ} - 4 0^{\circ} + \frac{36 0 ^{\circ} n}{1}

Least Common Multiplier of

2, 9, 1 : 18

2, 9, 1

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

2, 9, 1

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

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

18

For

9 0^{\circ} :

multiply the denominator and numerator by

9

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

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

For

\frac{36 0 ^{\circ} n}{1} :

multiply the denominator and numerator by

18

\frac{36 0 ^{\circ} n}{1} = \frac{36 0 ^{\circ} n \cdot 18}{1 \cdot 18} = \frac{648 0 ^{\circ} n}{18}

= 9 0^{\circ} - 4 0^{\circ} + \frac{648 0 ^{\circ} n}{18}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 9 - 72 0 ^{\circ} + 648 0 ^{\circ} n}{18}

Add similar elements:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}

= \frac{\frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18}}{2}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}

2 x = 18 0^{\circ} - (9 0^{\circ} - 4 0^{\circ}) + 36 0^{\circ} n : x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} - \frac{9 0 ^{\circ} - 4 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} : \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

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

Apply rule

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

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

Join

9 0^{\circ} - 4 0^{\circ} : 5 0^{\circ}

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

Least Common Multiplier of

2, 9 : 18

2, 9

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

2

9

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

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

18

For

9 0^{\circ} :

multiply the denominator and numerator by

9

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

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 0^{\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} 9 - 72 0 ^{\circ}}{18}

Add similar elements:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= 5 0^{\circ}

= \frac{18 0 ^{\circ} - 5 0 ^{\circ} + 36 0 ^{\circ} n}{2}

Join

18 0^{\circ} - 5 0^{\circ} + 36 0^{\circ} n : \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{18}

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

Convert element to fraction:

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

= 18 0^{\circ} - 5 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 18}{18}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 18 - 90 0 ^{\circ} + 36 0 ^{\circ} n \cdot 18}{18}

18 0^{\circ} 18 - 90 0^{\circ} + 36 0^{\circ} n \cdot 18 = 234 0^{\circ} + 648 0^{\circ} n

18 0^{\circ} 18 - 90 0^{\circ} + 36 0^{\circ} n \cdot 18

Add similar elements:

324 0^{\circ} - 90 0^{\circ} = 234 0^{\circ}

= 234 0^{\circ} + 2 \cdot 324 0^{\circ} n

Multiply the numbers:

2 \cdot 18 = 36

= 234 0^{\circ} + 648 0^{\circ} n

= \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{18}

= \frac{\frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{18}}{2}

Apply the fraction rule:

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

= \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

x = \frac{90 0 ^{\circ} + 648 0 ^{\circ} n}{36}, x = \frac{234 0 ^{\circ} + 648 0 ^{\circ} n}{36}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=cos(40) ?
The general solution for sin(2x)=cos(40) is x=(900+6480n)/(36),x=(2340+6480n)/(36)