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
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