# Thread: Eliminating parameters of parametric equations with respect to sin/cos/etc?

1. ## Eliminating parameters of parametric equations with respect to sin/cos/etc?

Hi,

In reading many tutorials on parametric equations, there is always a step that appears to be magic to me.

Take, for example, the parametric equation of x = sin(t), y = cos(t), 0 <= t <= pi.

To eliminate t, it is always referenced that x^2 + y^2 = 1, and so this is the final result. But that seems to be completely missing any steps.

I don't understand how the jump from question to result has occurred, and why that train of thought is taken. My intuition would just be to say t is arcsin(x), which makes for the messy equation of y = cos(arcsin(x)).

2. Originally Posted by charleschafsky
Hi,

In reading many tutorials on parametric equations, there is always a step that appears to be magic to me.

Take, for example, the parametric equation of x = sin(t), y = cos(t), 0 <= t <= pi.

To eliminate t, it is always referenced that x^2 + y^2 = 1, and so this is the final result. But that seems to be completely missing any steps.

I don't understand how the jump from question to result has occurred, and why that train of thought is taken. My intuition would just be to say t is arcsin(x), which makes for the messy equation of y = cos(arcsin(x)).
are you familar with the basic Pythagorean identity $\displaystyle \sin^2{t} + \cos^2{t} = 1$ ?

3. Yes, but normally when you're trying to eliminate parameters, you set one of the equations in terms of t, and then sub t in for the other. This case seems to be completely different. Is it generally assumed to use this identity when dealing with trigonemetric functions?

4. Originally Posted by charleschafsky
Yes, but normally when you're trying to eliminate parameters, you set one of the equations in terms of t, and then sub t in for the other. This case seems to be completely different. Is it generally assumed to use this identity when dealing with trigonemetric functions?
not all the time ... but it is most often used when sine and cosine are both involved.