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

# Question Video: Differentiating Functions Involving Reciprocal Trigonometric Ratios Using the Product Rule Mathematics • Higher Education

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

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### Video Transcript

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

Remember, the rectangular form of an equation is one which contains the variables π₯ and π¦ only. So weβll need to find a way to eliminate the third variable π‘ from our parametric equations. And at first glance, it doesnβt seem to be a nice way to do so. But letβs recall some trigonometric identities. We know that cos squared π plus sin squared π is equal to one. So letβs begin by simply squaring our expressions for π₯ and π¦.

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

Then, if we replace π with π‘ in our identity, remember that doesnβt change the identity. We see that we can replace cos squared π‘ with π₯ squared over nine. We can replace sin squared π‘ with π¦ 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 π¦ squared equals nine.

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