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

Convert the parametric equations π‘₯ = 3 cos 𝑑 and 𝑦 = 3 sin 𝑑 to rectangular form.

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Video Transcript

Convert the parametric equations π‘₯ equals three cos 𝑑 and 𝑦 equals three sin 𝑑 to rectangular form.

Remember, the rectangular form of an equation is one which contains the variables π‘₯ and 𝑦 only. So we’ll need to find a way to eliminate the third variable 𝑑 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 πœƒ plus sin squared πœƒ is equal to one. So let’s begin by simply squaring our expressions for π‘₯ and 𝑦.

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

Then, if we replace πœƒ with 𝑑 in our identity, remember that doesn’t change the identity. We see that we can replace cos squared 𝑑 with π‘₯ squared over nine. We can replace sin squared 𝑑 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 π‘₯ squared plus 𝑦 squared equals nine.

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