### Video Transcript

Convert the rectangular equation π¦ is equal to four to the polar form.

The question gives us a rectangular or a Cartesian equation π¦ is equal to four and wants us to convert this into an equivalent polar equation. What this means is we want to start with our Cartesian equation β in this case, π¦ is equal to four β eliminate the variables π₯ and π¦ and end up with an equivalent equation in terms of π and π.

To do this, we recall the standard polar relations. π₯ is equal to π multiplied by the cos of π and π¦ is equal to π multiplied by the sin of π. We can use these to substitute π¦ is equal to π sin π into our Cartesian equation to eliminate the variable π¦. Substituting π¦ is equal to π multiplied by the sin of π gives us that π sin π is equal to four.

Now, we could stop here. However, just like Cartesian equations β itβs standard to write them in the form π¦ equals some function of π₯ β in polar equations, itβs standard to try and write our equations as π is equal to some function of π. So, weβll divide both sides of our equation by the sin of π to give us an equation for π in terms of π. On the left-hand side of our equation, the shared factor of the sin of π in the numerator and the denominator cancel to give us π. And on the right-hand side of our equation, we have four divided by the sin of π.

Again, we could stop here now that we have π as a function of π. However, thereβs one more simplification we can do. We recall that one divided by the sin of π is equivalent to the csc of π. We then rearrange four divided by the sin of π to be four times one over the sin of π. Then, we substitute in that one divided by the sin of π is equivalent to the csc of π to give us π is equal to four multiplied by the csc of π. Therefore, what weβve shown is the rectangular equation π¦ is equal to four is equivalent to the polar equation π is equal to four multiplied by the csc of π.