I have to find the parametric functions that represent the curve:
$$\left(\frac{x – x_0}a\right)^2 + \left(\frac{y – y_0}b\right)^2 = 1$$
The notes simplify this to
$$\frac{(x – x_0)^2}{a^2} + \frac{(y – y_0)^2}{b^2} = 1$$
and then jump to saying that since $\cos^2t + \sin^2t = 1$,
$$\frac{x – x_0}a = \cos t\text{ and }\frac{y – y_0}b = \sin t$$
Where did the $t$ come from? and how is $\cos^2t + \sin^2t = 1$? I know how the $\cos$ and $\sin$ functions look, but im not sure how they got this formula and where they got $t$ from.
Best Answer
This is arguably the most important trigonometric identity --- the Pythagorean trigonometric identity.