### Video Transcript

Convert the equation π₯ squared
plus π¦ squared equals 25 into polar form.

Remember, we convert polar
coordinates to Cartesian or rectangular coordinates using the formulae π₯ equals π
cos π and π¦ equals π sin π. And these are suitable for all
values of π and π. In our original equation, weβve got
π₯ squared and π¦ squared. So letβs use our formulae for π₯
and π¦ to generate expressions for π₯ squared and π¦ squared in terms of π and
π.

Since π₯ is equal to π cos π, it
follows that π₯ squared must be π cos π all squared, which we can distribute and
say that π₯ squared is equal to π squared times cos squared π. Similarly, we see that π¦ squared
must be equal to π sin π all squared, which is equal to π squared sin squared
π.

Now our original equation says that
the sum of these is equal to 25. So we can say that π squared cos
squared π plus π squared sin squared π equals 25. Our next step is to factor π
squared on the left-hand side of this equation. So π squared times cos squared π
plus sin squared π equals 25. But why did we do this?

Well, here is where itβs useful to
know some of our trigonometric identities by heart. We know that cos squared π plus
sin squared π is equal to one for all values of π. So we can replace cos squared π
plus sin squared π in our equation with one. So π squared times one equals
25. Well, we donβt need this one. π squared is simply equal to
25. We solve this equation by taking
the square root of both sides. And we find that π is equal to
five.

Remember, we would usually take
both the positive and negative square root of 25. But since π represents a length,
we donβt need to. π₯ squared plus π¦ squared equals
25 is the same as π equals five in polar form.