What is the general solution for 6/(sin(60))= 4/(sin(x)) ?

The general solution for 6/(sin(60))= 4/(sin(x)) is x=0.61547…+360n,x=180-0.61547…+360n

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6/(sin(60))= 4/(sin(x))

Solution

\frac{6}{sin ( 6 0 ^{\circ} )} = \frac{4}{sin ( x )}

Solution

x = 0.61547 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

Radians

x = 0.61547 \dots + 2 πn, x = π - 0.61547 \dots + 2 πn

Solution steps

\frac{6}{sin ( 6 0 ^{\circ} )} = \frac{4}{sin ( x )}

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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{6}{\frac{3}{2}} = \frac{4}{sin ( x )}

Cross multiply

\frac{6}{\frac{3}{2}} = \frac{4}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

6 sin (x) = \frac{3}{2} \cdot 4

Simplify

\frac{3}{2} \cdot 4 : 23

\frac{3}{2} \cdot 4

Multiply fractions:

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

= \frac{3 \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 23

6 sin (x) = 23

6 sin (x) = 23

Divide both sides by

6

6 sin (x) = 23

Divide both sides by

6

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

Simplify

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

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

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{4}{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{3}{3}

Apply trig inverse properties

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

General solutions for

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

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

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

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

Show solutions in decimal form

x = 0.61547 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.61547 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 6/(sin(60))= 4/(sin(x)) ?
The general solution for 6/(sin(60))= 4/(sin(x)) is x=0.61547…+360n,x=180-0.61547…+360n