What is the general solution for (sin(x))/2+(sin(x))/3 = 1/4 ?

The general solution for (sin(x))/2+(sin(x))/3 = 1/4 is x=0.30469…+2pin,x=pi-0.30469…+2pin

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(sin(x))/2+(sin(x))/3 = 1/4

Solution

\frac{sin ( x )}{2} + \frac{sin ( x )}{3} = \frac{1}{4}

Solution

x = 0.30469 \dots + 2 πn, x = π - 0.30469 \dots + 2 πn

Degrees

x = 17.45760 \dots^{\circ} + 36 0^{\circ} n, x = 162.54239 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sin ( x )}{2} + \frac{sin ( x )}{3} = \frac{1}{4}

Simplify

\frac{sin ( x )}{2} + \frac{sin ( x )}{3} : \frac{5 sin ( x )}{6}

\frac{sin ( x )}{2} + \frac{sin ( x )}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{sin ( x )}{2} :

multiply the denominator and numerator by

3

\frac{sin ( x )}{2} = \frac{sin ( x ) \cdot 3}{2 \cdot 3} = \frac{sin ( x ) \cdot 3}{6}

For

\frac{sin ( x )}{3} :

multiply the denominator and numerator by

2

\frac{sin ( x )}{3} = \frac{sin ( x ) \cdot 2}{3 \cdot 2} = \frac{sin ( x ) \cdot 2}{6}

= \frac{sin ( x ) \cdot 3}{6} + \frac{sin ( x ) \cdot 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 3 + sin ( x ) \cdot 2}{6}

Add similar elements:

3 sin (x) + 2 sin (x) = 5 sin (x)

= \frac{5 sin ( x )}{6}

\frac{5 sin ( x )}{6} = \frac{1}{4}

Multiply both sides by

6

\frac{5 sin ( x )}{6} = \frac{1}{4}

Multiply both sides by

6

\frac{6 \cdot 5 sin ( x )}{6} = \frac{1 \cdot 6}{4}

Simplify

\frac{6 \cdot 5 sin ( x )}{6} = \frac{1 \cdot 6}{4}

Simplify

\frac{6 \cdot 5 sin ( x )}{6} : 5 sin (x)

\frac{6 \cdot 5 sin ( x )}{6}

Multiply the numbers:

6 \cdot 5 = 30

= \frac{30 sin ( x )}{6}

Divide the numbers:

\frac{30}{6} = 5

= 5 sin (x)

Simplify

\frac{1 \cdot 6}{4} : \frac{3}{2}

\frac{1 \cdot 6}{4}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{4}

Cancel the common factor:

2

= \frac{3}{2}

5 sin (x) = \frac{3}{2}

5 sin (x) = \frac{3}{2}

5 sin (x) = \frac{3}{2}

Divide both sides by

5

5 sin (x) = \frac{3}{2}

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{\frac{3}{2}}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{\frac{3}{2}}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

\frac{5 sin ( x )}{5}

Divide the numbers:

\frac{5}{5} = 1

= sin (x)

Simplify

\frac{\frac{3}{2}}{5} : \frac{3}{10}

\frac{\frac{3}{2}}{5}

Apply the fraction rule:

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

= \frac{3}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{3}{10}

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

Apply trig inverse properties

sin (x) = \frac{3}{10}

General solutions for

sin (x) = \frac{3}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{3}{10}) + 2 πn, x = π - arcsin (\frac{3}{10}) + 2 πn

x = arcsin (\frac{3}{10}) + 2 πn, x = π - arcsin (\frac{3}{10}) + 2 πn

Show solutions in decimal form

x = 0.30469 \dots + 2 πn, x = π - 0.30469 \dots + 2 πn

Popular Examples

2cos(x)sin(x)-cos(x)=0

2 cos (x) sin (x) - cos (x) = 0

cos(θ)= 2/3

cos (θ) = \frac{2}{3}

0=cos(x)

0 = cos (x)

tan(θ)=3

tan (θ) = 3

sin(x)= 5/13

sin (x) = \frac{5}{13}

Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))/2+(sin(x))/3 = 1/4 ?
The general solution for (sin(x))/2+(sin(x))/3 = 1/4 is x=0.30469…+2pin,x=pi-0.30469…+2pin