What is the general solution for (10)/(sin(x))=(100)/(sin(90)) ?

The general solution for (10)/(sin(x))=(100)/(sin(90)) is x=0.10016…+360n,x=180-0.10016…+360n

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Popular Trigonometry >

(10)/(sin(x))=(100)/(sin(90))

Solution

\frac{10}{sin ( x )} = \frac{100}{sin ( 9 0 ^{\circ} )}

Solution

x = 0.10016 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.10016 \dots + 36 0^{\circ} n

Radians

x = 0.10016 \dots + 2 πn, x = π - 0.10016 \dots + 2 πn

Solution steps

\frac{10}{sin ( x )} = \frac{100}{sin ( 9 0 ^{\circ} )}

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 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}

= 1

= 1

\frac{10}{sin ( x )} = \frac{100}{1}

Cross multiply

\frac{10}{sin ( x )} = \frac{100}{1}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

10 \cdot 1 = sin (x) \cdot 100

Simplify

10 \cdot 1 : 10

10 \cdot 1

Multiply the numbers:

10 \cdot 1 = 10

= 10

10 = sin (x) \cdot 100

10 = sin (x) \cdot 100

Switch sides

sin (x) \cdot 100 = 10

Divide both sides by

100

sin (x) \cdot 100 = 10

Divide both sides by

100

\frac{sin ( x ) \cdot 100}{100} = \frac{10}{100}

Simplify

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

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

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{10}{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{1}{10}

Apply trig inverse properties

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

General solutions for

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

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

x = arcsin (\frac{1}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{1}{10}) + 36 0^{\circ} n

x = arcsin (\frac{1}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{1}{10}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.10016 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.10016 \dots + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (10)/(sin(x))=(100)/(sin(90)) ?
The general solution for (10)/(sin(x))=(100)/(sin(90)) is x=0.10016…+360n,x=180-0.10016…+360n