What is the general solution for (37)/(sin(x))=(30)/(sin(30)) ?

The general solution for (37)/(sin(x))=(30)/(sin(30)) is x=0.66450…+360n,x=180-0.66450…+360n

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Popular Trigonometry >

(37)/(sin(x))=(30)/(sin(30))

Solution

\frac{37}{sin ( x )} = \frac{30}{sin ( 3 0 ^{\circ} )}

Solution

x = 0.66450 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.66450 \dots + 36 0^{\circ} n

Radians

x = 0.66450 \dots + 2 πn, x = π - 0.66450 \dots + 2 πn

Solution steps

\frac{37}{sin ( x )} = \frac{30}{sin ( 3 0 ^{\circ} )}

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{37}{sin ( x )} = \frac{30}{\frac{1}{2}}

Cross multiply

\frac{37}{sin ( x )} = \frac{30}{\frac{1}{2}}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

37 \cdot \frac{1}{2} = sin (x) \cdot 30

Simplify

37 \cdot \frac{1}{2} : \frac{37}{2}

37 \cdot \frac{1}{2}

Convert element to fraction:

37 = \frac{37}{1}

= \frac{1}{2} \cdot \frac{37}{1}

Multiply fractions:

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

= \frac{37 \cdot 1}{1 \cdot 2}

Cancel the common factor:

1

= \frac{37}{2}

\frac{37}{2} = sin (x) \cdot 30

\frac{37}{2} = sin (x) \cdot 30

Switch sides

sin (x) \cdot 30 = \frac{37}{2}

Divide both sides by

30

sin (x) \cdot 30 = \frac{37}{2}

Divide both sides by

30

\frac{sin ( x ) \cdot 30}{30} = \frac{\frac{37}{2}}{30}

Simplify

\frac{sin ( x ) \cdot 30}{30} = \frac{\frac{37}{2}}{30}

Simplify

\frac{sin ( x ) \cdot 30}{30} : sin (x)

\frac{sin ( x ) \cdot 30}{30}

Divide the numbers:

\frac{30}{30} = 1

= sin (x)

Simplify

\frac{\frac{37}{2}}{30} : \frac{37}{60}

\frac{\frac{37}{2}}{30}

Apply the fraction rule:

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

= \frac{37}{2 \cdot 30}

Multiply the numbers:

2 \cdot 30 = 60

= \frac{37}{60}

sin (x) = \frac{37}{60}

sin (x) = \frac{37}{60}

sin (x) = \frac{37}{60}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{37}{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{37}{60}

Apply trig inverse properties

sin (x) = \frac{37}{60}

General solutions for

sin (x) = \frac{37}{60}

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

x = arcsin (\frac{37}{60}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{37}{60}) + 36 0^{\circ} n

x = arcsin (\frac{37}{60}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{37}{60}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.66450 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.66450 \dots + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (37)/(sin(x))=(30)/(sin(30)) ?
The general solution for (37)/(sin(x))=(30)/(sin(30)) is x=0.66450…+360n,x=180-0.66450…+360n