Show that the curve with parametric equations $ x = t \cos t $, $ y = t \sin t $, $ z = t $ lies on the cone $ z^2 = x^2 + y^2 $, and use this fact to help sketch the curve.
substitute the parametric equations into the cone equation
So we're gonna make use of the fact that we have X equals because sign t co sign of T that is y equals t sign of t NZ just equals t. So knowing that we can adjust our parametric equations and we know that Z squared is going to be equal to X squared plus y square, that is our cone equation, Then we can plug in the different values we have, which gives us t squared equals t co sign t squared plus t sign t squared. That's just plugging in our values of X, y and Z and putting in the t values. So then we know that this is just going to equal t squared equal to, um we can factor out a t squared once we do all this so we get t squared equals C squared times co sign square T plus sine square t We know that coastline square T plus sine squared T is always going to equal one So we have a t squared equals C squared. Um, and since both sides are equal, since this is a tautology or something, that's absolutely true. We know that the equation is true. so the curve will lie on the cone. We know that if Z equals K, then since this is a cone like this of Z is constant, then our traces are going to be circles like that. We know if y equals K, so why is constant? Then we would end up getting hyperbole, as so we let y equal constantly get hyperbole. And also, since it's symmetrical, we know that if we let X equal k, we will also get hi purple eyes. And since the Parametric equations trace how to spiral with decreasing radius, um, as he gets closer to zero, um, and then it's an increasing radius as T is greater than zero. So we see when t increases, we get the increasing radius like that.