What is the general solution for (sqrt(116))/(sin(90))= 4/(sin(x)) ?

The general solution for (sqrt(116))/(sin(90))= 4/(sin(x)) is x=0.38050…+360n,x=180-0.38050…+360n

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Popular Trigonometry >

(sqrt(116))/(sin(90))= 4/(sin(x))

Solution

\frac{116}{sin ( 9 0 ^{\circ} )} = \frac{4}{sin ( x )}

Solution

x = 0.38050 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.38050 \dots + 36 0^{\circ} n

Radians

x = 0.38050 \dots + 2 πn, x = π - 0.38050 \dots + 2 πn

Solution steps

\frac{116}{sin ( 9 0 ^{\circ} )} = \frac{4}{sin ( x )}

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{116}{1} = \frac{4}{sin ( x )}

Cross multiply

\frac{116}{1} = \frac{4}{sin ( x )}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

116 sin (x) = 1 \cdot 4

Simplify

116 sin (x) = 1 \cdot 4

Simplify

116 sin (x) : 229 sin (x)

116 sin (x)

116 = 229

116

Prime factorization of

116 : 2^{2} \cdot 29

116

116

divides by

2116 = 58 \cdot 2

= 2 \cdot 58

58

divides by

258 = 29 \cdot 2

= 2 \cdot 2 \cdot 29

2, 29

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 29

= 2^{2} \cdot 29

= 2^{2} \cdot 29

Apply radical rule:

= 29 2^{2}

Apply radical rule:

2^{2} = 2

= 229

= 229 sin (x)

Simplify

1 \cdot 4 : 4

1 \cdot 4

Multiply the numbers:

1 \cdot 4 = 4

= 4

229 sin (x) = 4

229 sin (x) = 4

229 sin (x) = 4

Divide both sides by

229

229 sin (x) = 4

Divide both sides by

229

\frac{2 29 sin ( x )}{2 29} = \frac{4}{2 29}

Simplify

\frac{2 29 sin ( x )}{2 29} = \frac{4}{2 29}

Simplify

\frac{2 29 sin ( x )}{2 29} : sin (x)

\frac{2 29 sin ( x )}{2 29}

Divide the numbers:

\frac{2}{2} = 1

= \frac{29 sin ( x )}{29}

Cancel the common factor:

29

= sin (x)

Simplify

\frac{4}{2 29} : \frac{2 29}{29}

\frac{4}{2 29}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2}{29}

Rationalize

\frac{2}{29} : \frac{2 29}{29}

\frac{2}{29}

Multiply by the conjugate

\frac{29}{29}

= \frac{2 29}{29 29}

2929 = 29

2929

Apply radical rule:

a a = a

2929 = 29

= 29

= \frac{2 29}{29}

= \frac{2 29}{29}

sin (x) = \frac{2 29}{29}

sin (x) = \frac{2 29}{29}

sin (x) = \frac{2 29}{29}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{4}{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 29}{29}

Apply trig inverse properties

sin (x) = \frac{2 29}{29}

General solutions for

sin (x) = \frac{2 29}{29}

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 29}{29}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 29}{29}) + 36 0^{\circ} n

x = arcsin (\frac{2 29}{29}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 29}{29}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.38050 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.38050 \dots + 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sqrt(116))/(sin(90))= 4/(sin(x)) ?
The general solution for (sqrt(116))/(sin(90))= 4/(sin(x)) is x=0.38050…+360n,x=180-0.38050…+360n