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suppose I have an equations of the following with two unknowns $A$ and $\theta$

$A\sin(x+\theta)=D$

I have two points $(E,F) (G,H)$ how do I go about solving this equation analytically. I can solve this equation by using least squares where I just plugin a few numbers and solve it iteratively.

I was thinking about using trig identities and breaking it down to $A\sin(x)\cos(\theta)+Acos(x)\sin(\theta)=D$

But I am kind of stuck at that point. Using derivative to solve the equation doesn't help since the form of the equation is still $\cos(x+\theta)$

  • 1
    [Does this answer your question?](http://math.stackexchange.com/questions/34500/calculate-x-if-y-a-cdot-sinbx-cd/34523#34523)2012-09-14
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    The A term in my equation is still inside the arcsin function. I am not sure how to get it out of the arcsin function.2012-09-14
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    I'm sorry, I thought you wanted to solve for $x$. Where do you use $(E,F)$ and $(G,H)$ by the way?2012-09-14
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    Here would be the form of my two equations: A*sin(E+θ)=F,A*sin(G+θ)=H. I am trying to solve for A, θ2012-09-14
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    OK, now I understand your problem. Your idea of splitting the $\sin$ is a good one. Call $x=A\cos(\theta)$ and $y=A\sin(\theta)$. You then have two linear equations in $x$ and $y$. Solve these. Then, compute $x^2+y^2$ and $y/x$. Do you see how to work from there?2012-09-14
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    Yes I do. Thank you so much.2012-09-14
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    Actually can you explain x^2 + y^2 part what would that be used for?2012-09-14
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    $x^2+y^2=A^2$. But it is optional. Once you have $\theta$, you can solve $A$ out of $x=A\cos\theta$ as well.2012-09-14

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