Question #b9406

1 Answer
Jan 10, 2018

Answer:

#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)#?


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...#


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)#