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Math Help - help w/ trig equation

  1. #1
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    help w/ trig equation

    would like to solve the equation for theta:

    z = b*cos(theta) + a*cos(theta)

    been a long time since i took trig, any help would be appreciated.
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: help w/ trig equation

    Quote Originally Posted by bmorestudent View Post
    z = b*cos(theta) + a*cos(theta)
    The first step is to factor:
    z = b~cos(\theta) + a~cos(\theta) = (b + a)~cos(\theta)

    Can you finish it from there?

    -Dan
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  3. #3
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    Re: help w/ trig equation

    Are you sure you didn't mean z= a cos(\theta)+ b sin(\theta)? That's a much more interesting question. As topsquark says, your equation is just a matter of combining those two " cos(\theta) terms.

    To solve z= a cos(\theta)+ b sin(\theta), use the identity sin(A+B)= sin(A) cos(B)+ cos(A) sin(B). That would be the same as the previous equation if a= sin(A) and b= cos(A). The problem with that is that we would have to have a^2+ b^2= sin^2(A)+ cos^2(A) which is not, in general, true. We can fix that factoring \sqrt{a^2+ b^2} out. That is, we write the equation as
    z= \sqrt{a^2+ b^2}\left(\frac{a}{\sqrt{a^2+ b^2}}cos(\theta)+ \frac{b}{\sqrt{a^2+b^2}} sin(\theta)\right) so that
    \frac{z}{\sqrt{a^2+b^2}}= \left(\frac{a}{\sqrt{a^2+ b^2}}cos(\theta)+ \frac{b}{\sqrt{a^2+b^2}} sin(\theta)\right)
    So that now \left(\frac{a}{\sqrt{a^2+ b^2}}\right)^2+ \left(\frac{b}{\sqrt{a^2+ b^2}\right)^2}= \frac{a^2+ b^2}{a^2+ b^2}= 1

    Now, we can let \frac{a}{\sqrt{a^2+ b^2}}= sin(\phi) and \frac{b}{\sqrt{a^2+b^2}}= cos(\phi) so that the equation can be written
    sin(\phi)cos(\theta)+ cos(\phi)sin(\theta)= sin(\theta+ \phi)= \frac{z}{\sqrt{a^2+ b^2}}
    then \theta+ \phi= arcsin\left(\frac{z}{\sqrt{a^2+ b^2}}\right)
    \theta= arcsin\left(\frac{z}{\sqrt{a^2+ b^2}}\right)- \phi= arcsin\left(\frac{z}{\sqrt{a^2+ b^2}}\right)- arcsin(\frac{a}{\sqrt{a^2+ b^2}})
    Last edited by HallsofIvy; April 14th 2013 at 05:14 PM.
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    Re: help w/ trig equation

    thanks for the quick replies! yes, i meant to type b*SIN(theta), not cos. glad you caught that mistake and answered it hallsofivy, many thanks!
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