Find a Cartesian equation for the parametric equations: $$\begin{align} x&=\cos(t) \\ y&=7\sin(t) \end{align}$$
My attempt to solve the question was as follows: if we take the $\cos^{-1}(x)$ we can figure out what our $t$ was (we restrict the $t$ to $0\le t\le\pi$). Now we simply take the sine of this angle. Using the Pythagorean theorem, we can simplify $y=7\sin(\cos^{-1}(t))$ to $y=7\sqrt{1-x^2}$. On the other hand, the answer key says that the correct solution is $49x^2+y^2=49$.
I can see how they got to this answer (multiply first equation by $7$, square, and then add), but I don't understand two things. Why is my answer incorrect? Also, I understand that we can manipulate these equations and get the right answer, but what is really going on here (i.e. why do $x=\cos(t)$ and $y=7\sin(t)$ pop out the equation for an ellipse)?
Thanks!
Your answer is not completely incorrect, it is just incomplete. As you can see, a square root has two solutions to it: the positive and negative ones. You happened to express the positive one only (in this case, you only expressed the equation that describes the opper half of the curve). This is also due to the fact that you used a specific range for $t$, but it is not the only one you can use. In the end, it should look something like this: $$ y= \pm 7\sqrt{1 - x^2} $$ Which means, of course, that the points $(y, x)$ of the curve satisfy: $$ y= 7\sqrt{1 - x^2} \quad \text{or} \quad y= - 7\sqrt{1 - x^2}$$ To turn it into a compact form, you can square both sides of one of the equations and rearrange. $$ y^2 + 49x^2 = 49 $$