What is the general solution for (12)/(sin(120))= 5/(sin(x)) ?

The general solution for (12)/(sin(120))= 5/(sin(x)) is x=0.36917…+360n,x=180-0.36917…+360n

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(12)/(sin(120))= 5/(sin(x))

Solution

\frac{12}{sin ( 12 0 ^{\circ} )} = \frac{5}{sin ( x )}

Solution

x = 0.36917 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.36917 \dots + 36 0^{\circ} n

Radians

x = 0.36917 \dots + 2 πn, x = π - 0.36917 \dots + 2 πn

Solution steps

\frac{12}{sin ( 12 0 ^{\circ} )} = \frac{5}{sin ( x )}

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{3}{2}

\frac{12}{\frac{3}{2}} = \frac{5}{sin ( x )}

Cross multiply

\frac{12}{\frac{3}{2}} = \frac{5}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

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

Simplify

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

\frac{3}{2} \cdot 5

Multiply fractions:

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

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

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

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

Divide both sides by

12

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

Divide both sides by

12

\frac{12 sin ( x )}{12} = \frac{\frac{5 3}{2}}{12}

Simplify

\frac{12 sin ( x )}{12} = \frac{\frac{5 3}{2}}{12}

Simplify

\frac{12 sin ( x )}{12} : sin (x)

\frac{12 sin ( x )}{12}

Divide the numbers:

\frac{12}{12} = 1

= sin (x)

Simplify

\frac{\frac{5 3}{2}}{12} : \frac{5 3}{24}

\frac{\frac{5 3}{2}}{12}

Apply the fraction rule:

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

= \frac{5 3}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{5 3}{24}

sin (x) = \frac{5 3}{24}

sin (x) = \frac{5 3}{24}

sin (x) = \frac{5 3}{24}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{5}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{5 3}{24}

Apply trig inverse properties

sin (x) = \frac{5 3}{24}

General solutions for

sin (x) = \frac{5 3}{24}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{5 3}{24}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{5 3}{24}) + 36 0^{\circ} n

x = arcsin (\frac{5 3}{24}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{5 3}{24}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.36917 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.36917 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (12)/(sin(120))= 5/(sin(x)) ?
The general solution for (12)/(sin(120))= 5/(sin(x)) is x=0.36917…+360n,x=180-0.36917…+360n