Video Transcript
Convert the rectangular equation 𝑦 is equal to four to the polar form.
The question gives us a rectangular or a Cartesian equation 𝑦 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 — in this case, 𝑦 is equal to four — eliminate the variables 𝑥 and 𝑦 and end up with an equivalent equation in terms of 𝑟 and 𝜃.
To do this, we recall the standard polar relations. 𝑥 is equal to 𝑟 multiplied by the cos of 𝜃 and 𝑦 is equal to 𝑟 multiplied by the sin of 𝜃. We can use these to substitute 𝑦 is equal to 𝑟 sin 𝜃 into our Cartesian equation to eliminate the variable 𝑦. Substituting 𝑦 is equal to 𝑟 multiplied by the sin of 𝜃 gives us that 𝑟 sin 𝜃 is equal to four.
Now, we could stop here. However, just like Cartesian equations — it’s standard to write them in the form 𝑦 equals some function of 𝑥 — in polar equations, it’s standard to try and write our equations as 𝑟 is equal to some function of 𝜃. So, we’ll divide both sides of our equation by the sin of 𝜃 to give us an equation for 𝑟 in terms of 𝜃. On the left-hand side of our equation, the shared factor of the sin of 𝜃 in the numerator and the denominator cancel to give us 𝑟. And on the right-hand side of our equation, we have four divided by the sin of 𝜃.
Again, we could stop here now that we have 𝑟 as a function of 𝜃. However, there’s one more simplification we can do. We recall that one divided by the sin of 𝜃 is equivalent to the csc of 𝜃. We then rearrange four divided by the sin of 𝜃 to be four times one over the sin of 𝜃. Then, we substitute in that one divided by the sin of 𝜃 is equivalent to the csc of 𝜃 to give us 𝑟 is equal to four multiplied by the csc of 𝜃. Therefore, what we’ve shown is the rectangular equation 𝑦 is equal to four is equivalent to the polar equation 𝑟 is equal to four multiplied by the csc of 𝜃.
