What is the general solution for (50)/(sin(105))=(20)/(sin(x)) ?

The general solution for (50)/(sin(105))=(20)/(sin(x)) is x=0.39669…+360n,x=180-0.39669…+360n

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(50)/(sin(105))=(20)/(sin(x))

Solution

\frac{50}{sin ( 10 5 ^{\circ} )} = \frac{20}{sin ( x )}

Solution

x = 0.39669 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.39669 \dots + 36 0^{\circ} n

Radians

x = 0.39669 \dots + 2 πn, x = π - 0.39669 \dots + 2 πn

Solution steps

\frac{50}{sin ( 10 5 ^{\circ} )} = \frac{20}{sin ( x )}

sin (10 5^{\circ}) = \frac{2 ( 3 + 1 )}{4}

sin (10 5^{\circ})

Rewrite using trig identities:

sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

sin (10 5^{\circ})

Write

sin (10 5^{\circ})

sin (6 0^{\circ} + 4 5^{\circ})

= sin (6 0^{\circ} + 4 5^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\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}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\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{2}{2}

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

Simplify

\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2} : \frac{2 ( 3 + 1 )}{4}

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

Factor out common term

\frac{2}{2}

= \frac{2}{2} (\frac{3}{2} + \frac{1}{2})

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

\frac{3}{2} + \frac{1}{2}

Apply rule

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

= \frac{3 + 1}{2}

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

Multiply fractions:

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

= \frac{( 3 + 1 ) 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( 1 + 3 )}{4}

= \frac{2 ( 3 + 1 )}{4}

\frac{50}{\frac{2 ( 3 + 1 )}{4}} = \frac{20}{sin ( x )}

Cross multiply

\frac{50}{\frac{2 ( 3 + 1 )}{4}} = \frac{20}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

50 sin (x) = \frac{2 ( 3 + 1 )}{4} \cdot 20

Simplify

\frac{2 ( 3 + 1 )}{4} \cdot 20 : 52 (1 + 3)

\frac{2 ( 3 + 1 )}{4} \cdot 20

Multiply fractions:

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

= \frac{2 ( 3 + 1 ) \cdot 20}{4}

Divide the numbers:

\frac{20}{4} = 5

= 52 (1 + 3)

50 sin (x) = 52 (1 + 3)

50 sin (x) = 52 (1 + 3)

Divide both sides by

50

50 sin (x) = 52 (1 + 3)

Divide both sides by

50

\frac{50 sin ( x )}{50} = \frac{5 2 ( 1 + 3 )}{50}

Simplify

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

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

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{20}{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{2 ( 1 + 3 )}{10}

Apply trig inverse properties

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

General solutions for

sin (x) = \frac{2 ( 1 + 3 )}{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{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n

x = arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.39669 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.39669 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (50)/(sin(105))=(20)/(sin(x)) ?
The general solution for (50)/(sin(105))=(20)/(sin(x)) is x=0.39669…+360n,x=180-0.39669…+360n