Question

Question #b9406

Answer 1

`sin(8theta)=u/(8a)`

Explanation:

I'm not too sure of your question.

I take it you are asking:

`sintheta=u/a`; What is `sin(8theta)`?

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We can relate this back to transformations of graphs in general. We're going to be moving away from trigonometry for just a bit!

For example:

`"Let " f(x)=x^2-9x+18`
How about for a moment we consider the roots of `f(x)=0`
`"Let " f(x)=0`
`x^2-9x+18=0`
`(x-3)(x-6)=0`
So this equation has roots of `x=3` and `x=6`
graph{y=x^2-9x+18 [-10, 10, -5, 5]}

Consider `f(2x)`
`f(2x)=(2x)^2-9(2x)+18`
`=4x^2-18x+18`
Now, consider the roots of `f(2x)=0`
`"Let " f(2x)=0`
`4x^2-18x+18=0`
`2x^2-9x+9=0`
`x=(9+-sqrt((-9)^2-4xx2xx9))/(2xx2)`
`x=3` or `x=3/2`
graph{4x^2-18x+18 [-10, 10, -5, 5]}

Look at the roots.
`f(x)` has roots `x=6` and `x=3`
`f(2x)` has roos `x=6/2` and `x=3/2`

This isn't a proof, but it's enough for us to infer that:

if an expression `f(x)` has roots `x=alpha, beta, gamma, delta...`, then `f(kx)` has roots `x=alpha/k, beta/k, gamma/k, delta/k...`

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Back to our trig

`"Let " g(x)=sinx` (because I used `f(x)` further up).

For some value `x=theta, g(theta)=u/a`
`g(8x)=sin(8x)`
`sin(8theta)=g(8theta)=(u/a)/8`
`sin(8theta)=u/(8a)`