What is the general solution for (sin(82))/(sin(x))=sqrt(5) ?

The general solution for (sin(82))/(sin(x))=sqrt(5) is x=0.45878…+360n,x=180-0.45878…+360n

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(sin(82))/(sin(x))=sqrt(5)

Solution

\frac{sin ( 8 2 ^{\circ} )}{sin ( x )} = 5

Solution

x = 0.45878 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.45878 \dots + 36 0^{\circ} n

Radians

x = 0.45878 \dots + 2 πn, x = π - 0.45878 \dots + 2 πn

Solution steps

\frac{sin ( 8 2 ^{\circ} )}{sin ( x )} = 5

Multiply both sides by

sin (x)

\frac{sin ( 8 2 ^{\circ} )}{sin ( x )} = 5

Multiply both sides by

sin (x)

\frac{sin ( 8 2 ^{\circ} )}{sin ( x )} sin (x) = 5 sin (x)

Simplify

sin (8 2^{\circ}) = 5 sin (x)

sin (8 2^{\circ}) = 5 sin (x)

Switch sides

5 sin (x) = sin (8 2^{\circ})

Divide both sides by

5

5 sin (x) = sin (8 2^{\circ})

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{sin ( 8 2 ^{\circ} )}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{sin ( 8 2 ^{\circ} )}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

\frac{5 sin ( x )}{5}

Cancel the common factor:

5

= sin (x)

Simplify

\frac{sin ( 8 2 ^{\circ} )}{5} : \frac{5 sin ( 8 2 ^{\circ} )}{5}

\frac{sin ( 8 2 ^{\circ} )}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{sin ( 8 2 ^{\circ} ) 5}{5 5}

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5 sin ( 8 2 ^{\circ} )}{5}

sin (x) = \frac{5 sin ( 8 2 ^{\circ} )}{5}

sin (x) = \frac{5 sin ( 8 2 ^{\circ} )}{5}

sin (x) = \frac{5 sin ( 8 2 ^{\circ} )}{5}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{sin ( 8 2 ^{\circ} )}{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{5 sin ( 8 2 ^{\circ} )}{5}

Apply trig inverse properties

sin (x) = \frac{5 sin ( 8 2 ^{\circ} )}{5}

General solutions for

sin (x) = \frac{5 sin ( 8 2 ^{\circ} )}{5}

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

x = arcsin (\frac{5 sin ( 8 2 ^{\circ} )}{5}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{5 sin ( 8 2 ^{\circ} )}{5}) + 36 0^{\circ} n

x = arcsin (\frac{5 sin ( 8 2 ^{\circ} )}{5}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{5 sin ( 8 2 ^{\circ} )}{5}) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.45878 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.45878 \dots + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(82))/(sin(x))=sqrt(5) ?
The general solution for (sin(82))/(sin(x))=sqrt(5) is x=0.45878…+360n,x=180-0.45878…+360n