Video: Differentiating Functions Involving Reciprocal Trigonometric Ratios Using the Product Rule

Video Transcript

Convert the parametric equations 𝑥 equals three cos 𝑡 and 𝑦 equals three sin 𝑡 to rectangular form.

Remember, the rectangular form of an equation is one which contains the variables 𝑥 and 𝑦 only. So we’ll need to find a way to eliminate the third variable 𝑡 from our parametric equations. And at first glance, it doesn’t seem to be a nice way to do so. But let’s recall some trigonometric identities. We know that cos squared 𝜃 plus sin squared 𝜃 is equal to one. So let’s begin by simply squaring our expressions for 𝑥 and 𝑦.

With 𝑥, we get 𝑥 squared equals three cos 𝑡 all squared, which is equal to nine cos squared 𝑡. And so, we can say that cos squared 𝑡 must be equal to 𝑥 squared over nine. Similarly, we can say that 𝑦 squared is equal to three sin 𝑡 all squared. We distribute the parentheses and we find that 𝑦 squared is nine times sin squared 𝑡. And then, we divide through by nine. And we see that sin squared 𝑡 is equal to 𝑦 squared over nine.

Then, if we replace 𝜃 with 𝑡 in our identity, remember that doesn’t change the identity. We see that we can replace cos squared 𝑡 with 𝑥 squared over nine. We can replace sin squared 𝑡 with 𝑦 squared over nine. And then, this is all equal to one. We next multiply through by nine. And we find that the rectangular form of our equation is equal to 𝑥 squared plus 𝑦 squared equals nine.