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New here and looking for assistance.

I'm proposing a problem where there is a sinusoid with the equation $y=4\sin(0.5x-1)$. This will intercept a circle at $(x-4)^2 + (y-3)^2 = 4$.

I'm wondering how to go about solving such a beast. I know that I must substitute my sinusoid in for $y$ in my circle, but when I multiply out my brackets, I end up with a whole load of mess that I'm absolutely clueless to clean up.

I know from both graphing it on Desmos and computing the substituted equation on Wolfram Alpha that the $x$-intercepts are at $2.75$ and $5.855$ but I have no idea how to solve that out of my substituted equation. It looks like I end up with a double quadratic with both powers of $x$ and powers of sine!

Any tips?

Many thanks! Chris

Harry Peter
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C. Wolfe
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  • I think that with standard equating technique you're just going to end up with some mess. I would translate across to make the function inside the $sin$ a little neater (move 1 to the left). The only other idea I have is to use some kind of iterative technique like Newton-Raphson to get arbitrarily close. Good question. – Leonhard Euler Jun 18 '18 at 14:06
  • So basically, if I were to ask someone to solve this, I'd really better let them use the computer to crunch the numbers then. :) – C. Wolfe Jun 18 '18 at 14:24
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    Yes, I'd say so. – Leonhard Euler Jun 18 '18 at 15:19

1 Answers1

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We find maximum and minimum of functions:

For sinusoide:

$y=4 \sin (0.5 x -1) \implies y' = 0.5 \cos (0.5 x -1)= 0 \implies x/2 -1= \pi/2 \implies x_s=\pi +2$ and $y_s =4$ is maximum point. Also sinusoide cross x axis at points $2, 0$ and $(x = 2\pi +1 ≈ 7.28, 0) $

For circle maximum is $(x_c, y_c)= (4, 5)$ and minimum is $(x_c, y_c)=(4, 1)$

We can see that $2 < x_c < 7.28$ and $ y_s=4< y_c$ this indicates that two graphs intersect on two points; first point is $2<x_1<x_c=4 $ and $1 < y_1< y_c= 5$. Second point is $4 < x_2 < 6$ where $x=6$ is right extremum of circle and $1< y_2 < y_c=5 $. By try and error we can estimate the coordinates of intersection points

CiaPan
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sirous
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  • If you prefix the trigonometric functions' names with a backslash, LaTeX/MathJax will recognize them as functions' symbols and accordingly render them in upright font with proper spacing: \sin x → $\sin x$ instead of a blob of letters sin x → $sin x$ which looks like a product $s\cdot i\cdot n\cdot x$. – CiaPan Jun 25 '18 at 11:43
  • Thank you CiaPan for advise. – sirous Jun 25 '18 at 13:55