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

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

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

Solution

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

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

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

Multiply both sides by

10

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

Multiply both sides by

10

\frac{sin ( x )}{37} \cdot 10 = \frac{\frac{1}{2}}{30} \cdot 10

Simplify

\frac{sin ( x )}{37} \cdot 10 = \frac{\frac{1}{2}}{30} \cdot 10

Simplify

\frac{sin ( x )}{37} \cdot 10 : \frac{10 sin ( x )}{37}

\frac{sin ( x )}{37} \cdot 10

Multiply fractions:

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

= \frac{sin ( x ) \cdot 10}{37}

Simplify

\frac{\frac{1}{2}}{30} \cdot 10 : \frac{1}{6}

\frac{\frac{1}{2}}{30} \cdot 10

\frac{\frac{1}{2}}{30} = \frac{1}{60}

\frac{\frac{1}{2}}{30}

Apply the fraction rule:

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

= \frac{1}{2 \cdot 30}

Multiply the numbers:

2 \cdot 30 = 60

= \frac{1}{60}

= 10 \cdot \frac{1}{60}

Multiply fractions:

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

= \frac{1 \cdot 10}{60}

Multiply the numbers:

1 \cdot 10 = 10

= \frac{10}{60}

Cancel the common factor:

10

= \frac{1}{6}

\frac{10 sin ( x )}{37} = \frac{1}{6}

\frac{10 sin ( x )}{37} = \frac{1}{6}

\frac{10 sin ( x )}{37} = \frac{1}{6}

Multiply both sides by

37

\frac{10 sin ( x )}{37} = \frac{1}{6}

Multiply both sides by

37

\frac{37 \cdot 10 sin ( x )}{37} = \frac{1 \cdot 37}{6}

Simplify

\frac{37 \cdot 10 sin ( x )}{37} = \frac{1 \cdot 37}{6}

Simplify

\frac{37 \cdot 10 sin ( x )}{37} : 10 sin (x)

\frac{37 \cdot 10 sin ( x )}{37}

Multiply the numbers:

37 \cdot 10 = 370

= \frac{370 sin ( x )}{37}

Divide the numbers:

\frac{370}{37} = 10

= 10 sin (x)

Simplify

\frac{1 \cdot 37}{6} : \frac{37}{6}

\frac{1 \cdot 37}{6}

Multiply the numbers:

1 \cdot 37 = 37

= \frac{37}{6}

10 sin (x) = \frac{37}{6}

10 sin (x) = \frac{37}{6}

10 sin (x) = \frac{37}{6}

Divide both sides by

10

10 sin (x) = \frac{37}{6}

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{\frac{37}{6}}{10}

Simplify

\frac{10 sin ( x )}{10} = \frac{\frac{37}{6}}{10}

Simplify

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

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

Divide the numbers:

\frac{10}{10} = 1

= sin (x)

Simplify

\frac{\frac{37}{6}}{10} : \frac{37}{60}

\frac{\frac{37}{6}}{10}

Apply the fraction rule:

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

= \frac{37}{6 \cdot 10}

Multiply the numbers:

6 \cdot 10 = 60

= \frac{37}{60}

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

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

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

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Frequently Asked Questions (FAQ)

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