Convert the parametric equations 𝑥 is equal to the cos of 𝑡 and 𝑦 is equal to the sin of 𝑡 to rectangular form.
We’re given a pair of parametric equations, and we’re asked to convert this into the rectangular form. Remember, this means we need to rewrite this as an equation in terms of 𝑥 and 𝑦. We want to eliminate our variable 𝑡. The first thing we’ll do is look at our parametric equations. We can see we have 𝑥 is equal to the cos of 𝑡 and 𝑦 is equal to the sin of 𝑡. To eliminate the variable 𝑡, we’re going to want to find some kind of relationship between the cos of 𝑡 and the sin of 𝑡.
And luckily, we do know a relationship between these two functions. The Pythagorean theorem tells us the sin squared of 𝜃 plus the cos squared of 𝜃 will be equivalent to one for any value of 𝜃. And we used the equivalent sign here because this is true for any value 𝜃. However, you might see this written with an equal sign. Of course, it doesn’t matter what we label our variable in the Pythagorean theorem. However, because in the question we’re working with the cos of 𝑡 and the sin of 𝑡, we’ll relabel our variable 𝑡.
So now we have for any value 𝑡, the sin squared of 𝑡 plus the cos squared of 𝑡 is equal to one. Of course, we can then rewrite the sin squared of 𝑡 as the sin of 𝑡 all squared and the cos squared of 𝑡 as the cos of 𝑡 all squared. And the cos of 𝑡 is equal to 𝑥, and the sin of 𝑡 is equal to 𝑦. So what we need to do is replace the sin of 𝑡 with 𝑦 and the cos of 𝑡 with 𝑥. This gives us 𝑦 squared plus 𝑥 squared is equal to one. And it’s worth pointing out here this is no longer an identity. It’s now an equation. This is because it’s only true for certain pairs 𝑥 and 𝑦 whereas, before, our Pythagorean identity was true for any real value of 𝑡.
We could leave our answer like this. However, we’ll also reorder our 𝑥- and 𝑦-terms, giving us 𝑥 squared plus 𝑦 squared is equal to one. Therefore, we were able to convert the parametric equations 𝑥 is equal to the cos of 𝑡 and 𝑦 is equal to sin of 𝑡 into rectangular form. We got 𝑥 squared plus 𝑦 squared will be equal to one.