I Need Help Figuring Out This Equation.

I can't figure out how to solve this equations, can anybody help me?

This looks a lot like the Discrete Fourier Transform, at least in a special case where [math]f_p[/math] is proportional to [math]p.[/math]Let’s write your equations this way so we can compare it to the DFT. We get all [math]2n[/math] equations here in one line:[math]\displaystyle I_p + i Q_p = \sum_{q=1}^{n} A_q (\cos(k_q f_p) + i \sin(k_q f_p)) = \sum_q A_q e^{i k_q f_p}[/math]Compare this is the DFT below. I’ve altered the typical subscripts to match your equations:[math]\displaystyle X_p = \sum_{q=0}^{n-1} x_q e^{-2 \pi i p q/n}[/math]We don’t have to worry about the slightly different summing ranges. In the DFT [math]q=0[/math] and [math]q=n[/math] give the same term (if we cyclically set [math]x_n = x_0[/math]).If we make the identity, we have[math]X_p = I_p + i Q_p [/math][math]x_q = A_q[/math][math]-2 \pi i p q/n = i k_q f_p[/math]Solving for the unknown [math]k[/math]s:[math]k_q = - 2 \pi (p/f_p) q / n [/math]This only makes sense if [math]p/f_p[/math] is a constant, not changing with each [math]p.[/math] Let’s write [math]p/f_p = 1/f_1[/math] and suck up all the constants as [math]c= - 2\pi /(n f_1).[/math][math]k_q = c q[/math]What this says is in the special case where [math]f_p/p[/math] is constant, the [math]k[/math]s are multiples of a fundamental [math]c.[/math]To get the [math]A[/math]s, we see they’re identified with the time domain signal [math]x_q.[/math]In the problem statement we’re given the [math]X_p,[/math] corresponding to the frequency domain, i.e. the Fourier transform, of our time domain signal.So to recover the [math]A_p[/math] we need the IDFT, the Inverse Discrete Fourier Transform. It’s basically the same as the DFT; there’s a conjugate and a factor of [math]1/n[/math] different.[math]\displaystyle x_q = \dfrac 1 n \sum_{p=0}^{n-1} X_p e^{2 \pi i p q/n}[/math]Substituting[math]\displaystyle A_q = \dfrac 1 n \sum_{p=0}^{n-1} ( I_p + i Q_p ) e^{2 \pi i p q/n}[/math]If I really had to solve it in the arbitrary case, where the frequency samples aren’t evenly spaced, I think there’s an approach using the [math]z[/math] transform. The inverse [math]z[/math] transform is much more complicated than the IDFT, but that’s probably appropriate to the much more complicated question.

How do I figure out the polar equation of this graph?

Sorry for the delay in my response.As given, there is no unique equation for the line. Many lines could be drawn through the point, with varying slopes.If you assume that 4 is the "distance" of the line from the origin (that the line segment labeled to have a length of 4 is perpendicular to the line), then there's a unique solution.Here's what I came up with.  You might be able to simplify the expression for r(theta).  Hope I'm correct and hope it helps...The range of inputs (theta) for r(theta) will not be all values of theta, but be constrained to (-40, 140) degrees (when the r vector becomes parallel to the given line as r goes to infinity) (theta measured with respect to the x-axis, counter-clockwise opening set to be positive).I think the second form makes it more clear that r goes to infinity as theta goes to either -40 degrees or +140 degrees.