# Question Video: Converting Trigonometric Parametric Equations into Rectangular Form Mathematics • Higher Education

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

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### Video Transcript

Convert the parametric equations π₯ is equal to the cos of π‘ and π¦ is equal to the sin of π‘ to rectangular form.

Weβre given a pair of parametric equations, and weβre asked to convert this into the rectangular form. Remember, this means we need to rewrite this as an equation in terms of π₯ and π¦. We want to eliminate our variable π‘. The first thing weβll do is look at our parametric equations. We can see we have π₯ is equal to the cos of π‘ and π¦ is equal to the sin of π‘. To eliminate the variable π‘, weβre going to want to find some kind of relationship between the cos of π‘ and the sin of π‘.

And luckily, we do know a relationship between these two functions. The Pythagorean theorem tells us the sin squared of π plus the cos squared of π will be equivalent to one for any value of π. And we used the equivalent sign here because this is true for any value π. However, you might see this written with an equal sign. Of course, it doesnβt matter what we label our variable in the Pythagorean theorem. However, because in the question weβre working with the cos of π‘ and the sin of π‘, weβll relabel our variable π‘.

So now we have for any value π‘, the sin squared of π‘ plus the cos squared of π‘ is equal to one. Of course, we can then rewrite the sin squared of π‘ as the sin of π‘ all squared and the cos squared of π‘ as the cos of π‘ all squared. And the cos of π‘ is equal to π₯, and the sin of π‘ is equal to π¦. So what we need to do is replace the sin of π‘ with π¦ and the cos of π‘ with π₯. This gives us π¦ squared plus π₯ squared is equal to one. And itβs worth pointing out here this is no longer an identity. Itβs now an equation. This is because itβs only true for certain pairs π₯ and π¦ whereas, before, our Pythagorean identity was true for any real value of π‘.

We could leave our answer like this. However, weβll also reorder our π₯- and π¦-terms, giving us π₯ squared plus π¦ squared is equal to one. Therefore, we were able to convert the parametric equations π₯ is equal to the cos of π‘ and π¦ is equal to sin of π‘ into rectangular form. We got π₯ squared plus π¦ squared will be equal to one.