Question

PLEASE HELP!!! Solve 5sin(π/3x)=3 for the four smallest positive solutions

Answer 1

Let's solve ~

This one's a special case of a right angled triangle with sides (3, 4, and 5 units)

Back to the problem :

`\qquad\displaystyle \tt \dashrightarrow \: 5 \sin \bigg( ( \pi)/(3) x \bigg) = 3`

`\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( ( \pi)/(3) x \bigg) = (3)/(5)`

Now, check the triangle, sin 37° = 3/5

therefore,

`\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( ( \pi)/(3) x \bigg) = \sin(37 \degree)`

[ convert degrees on right side to radians ]

`\qquad\displaystyle \tt \dashrightarrow \: \sin \bigg( ( \pi)/(3) x \bigg) = \sin \bigg(37 \degree * ( \pi)/(180 \degree) \bigg )`

There are three more possible values as :

`\qquad\displaystyle \tt \dashrightarrow \: \sin( \theta) = \sin(\pi - \theta)`

`\qquad\displaystyle \tt \dashrightarrow \: sin( \theta) = \sin \bigg( { 2\pi}{} + \theta \bigg)`

`\qquad\displaystyle \tt \dashrightarrow \: sin( \theta) = \sin \bigg( \frac{ 3\pi}{} - \theta\bigg)`

Equating both, we get :

First value :

`\qquad\displaystyle \tt \dashrightarrow \: ( \pi)/(3) x = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: x = 37 * ( \cancel \pi)/(180) * (3)/( \cancel \pi)`

`\qquad\displaystyle \tt \dashrightarrow \: x = (37)/(60)`

or in decimals :

`\qquad\displaystyle \tt \dashrightarrow \: x = 0.616666... = 0.6167`

[ 6 repeats at third place after decimal, till four decimal places it would be 0.6167 after rounding off ]

similarly,

Second value :

`\qquad\displaystyle \tt \dashrightarrow \: \pi - ( \pi)/(3) x = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(1 - (x)/(3) \bigg ) = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: 1 - (x)/(3) = (37)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: - (x)/(3) = 0.205 - 1`

`\qquad\displaystyle \tt \dashrightarrow \: (x)/(3) = 0.795`

`\qquad\displaystyle \tt \dashrightarrow \: x = 3 * 0.795`

`\qquad\displaystyle \tt \dashrightarrow \: x = 2.385`

Third value :

`\qquad\displaystyle \tt \dashrightarrow \: 2\pi + ( \pi)/(3) x = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(2 + (x)/(3) \bigg ) = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: 2 + (x)/(3) = (37)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: (x)/(3) = 0.205 - 2`

`\qquad\displaystyle \tt \dashrightarrow \: (x)/(3) = - 1.795`

`\qquad\displaystyle \tt \dashrightarrow \: x = 3 * -1 .795`

`\qquad\displaystyle \tt \dashrightarrow \: x = -5.385`

Fourth value :

`\qquad\displaystyle \tt \dashrightarrow \: 3 \pi - ( \pi)/(3) x = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: \pi \bigg(3 - (x)/(3) \bigg ) = 37 * ( \pi)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: 3 - (x)/(3) = (37)/(180)`

`\qquad\displaystyle \tt \dashrightarrow \: - (x)/(3) = 0.205 - 3`

`\qquad\displaystyle \tt \dashrightarrow \: (x)/(3) = 2.795`

`\qquad\displaystyle \tt \dashrightarrow \: x = 3 * 2.795`

`\qquad\displaystyle \tt \dashrightarrow \: x = 8.385`

" x can have infinite number of values here with the same result, here are the four values as you requested "

I hope it was helpful ~