Video: Converting the Rectangular Equation into Polar Form

Convert the rectangular equation 𝑦 = 4 to the polar form.

01:59

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 β€” in this case, 𝑦 is equal to four β€” eliminate the variables π‘₯ and 𝑦 and end up with an equivalent equation in terms of π‘Ÿ and πœƒ.

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

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

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

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