What is the general solution for (10)/(sin(30))= 3/(sin(b)) ?

The general solution for (10)/(sin(30))= 3/(sin(b)) is b=0.15056…+360n,b=180-0.15056…+360n

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(10)/(sin(30))= 3/(sin(b))

Solution

\frac{10}{sin ( 3 0 ^{\circ} )} = \frac{3}{sin ( b )}

Solution

b = 0.15056 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.15056 \dots + 36 0^{\circ} n

Radians

b = 0.15056 \dots + 2 πn, b = π - 0.15056 \dots + 2 πn

Solution steps

\frac{10}{sin ( 3 0 ^{\circ} )} = \frac{3}{sin ( b )}

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

= \frac{1}{2}

\frac{10}{\frac{1}{2}} = \frac{3}{sin ( b )}

Cross multiply

\frac{10}{\frac{1}{2}} = \frac{3}{sin ( b )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

10 sin (b) = \frac{1}{2} \cdot 3

Simplify

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

\frac{1}{2} \cdot 3

Convert element to fraction:

3 = \frac{3}{1}

= \frac{1}{2} \cdot \frac{3}{1}

Multiply fractions:

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

= \frac{1 \cdot 3}{2 \cdot 1}

Cancel the common factor:

1

= \frac{3}{2}

10 sin (b) = \frac{3}{2}

10 sin (b) = \frac{3}{2}

Divide both sides by

10

10 sin (b) = \frac{3}{2}

Divide both sides by

10

\frac{10 sin ( b )}{10} = \frac{\frac{3}{2}}{10}

Simplify

\frac{10 sin ( b )}{10} = \frac{\frac{3}{2}}{10}

Simplify

\frac{10 sin ( b )}{10} : sin (b)

\frac{10 sin ( b )}{10}

Divide the numbers:

\frac{10}{10} = 1

= sin (b)

Simplify

\frac{\frac{3}{2}}{10} : \frac{3}{20}

\frac{\frac{3}{2}}{10}

Apply the fraction rule:

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

= \frac{3}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{3}{20}

sin (b) = \frac{3}{20}

sin (b) = \frac{3}{20}

sin (b) = \frac{3}{20}

Verify Solutions

Find undefined (singularity) points:

sin (b) = 0

Take the denominator(s) of

\frac{3}{sin ( b )}

and compare to zero

sin (b) = 0

The following points are undefined

sin (b) = 0

Combine undefined points with solutions:

sin (b) = \frac{3}{20}

Apply trig inverse properties

sin (b) = \frac{3}{20}

General solutions for

sin (b) = \frac{3}{20}

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

b = arcsin (\frac{3}{20}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{3}{20}) + 36 0^{\circ} n

b = arcsin (\frac{3}{20}) + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin (\frac{3}{20}) + 36 0^{\circ} n

Show solutions in decimal form

b = 0.15056 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.15056 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (10)/(sin(30))= 3/(sin(b)) ?
The general solution for (10)/(sin(30))= 3/(sin(b)) is b=0.15056…+360n,b=180-0.15056…+360n