Video Transcript
Consider the Cartesian equation 𝑦 is equal to two 𝑥 plus three. Complete the following steps to find the polar form of the equation by writing an equivalent equation each time. First, use the fact that 𝑥 is equal to 𝑟 multiplied by the cos of 𝜃 to eliminate 𝑥. Now, use the fact that 𝑦 is equal to 𝑟 sin 𝜃 to eliminate 𝑦. Finally, make 𝑟 the subject.
The question gives us the Cartesian equation 𝑦 is equal to two 𝑥 plus three. And it wants us to find the polar form of this equation. And it wants us to do this by using manipulations which leave us with an equivalent equation at every step. The first thing it wants us to do is use one of our standard polar relations, 𝑥 is equal to 𝑟 cos of 𝜃. It wants us to use this to eliminate 𝑥 from our Cartesian equation. So we’ll start with our Cartesian equation, 𝑦 is equal to two 𝑥 plus three. And we’re going to substitute 𝑥 is equal to 𝑟 cos of 𝜃 for every instance of 𝑥 in our Cartesian equation.
Substituting 𝑥 is equal to 𝑟 cos of 𝜃 gives us that 𝑦 is equal to two 𝑟 cos of 𝜃 plus three. So we’ve shown that 𝑦 is equal to two 𝑟 cos of 𝜃 plus three and that this is an equivalent equation to our original Cartesian equation. And we’ve eliminated the variable 𝑥. Now, the question wants us to use our other standard polar relation that 𝑦 is equal to 𝑟 sin 𝜃. It wants us to use this to eliminate the variable 𝑦. So we’ll start with the equation we got in the first part, 𝑦 is equal to two 𝑟 cos of 𝜃 plus three. And we’re gonna use the substitution, 𝑦 is equal to 𝑟 sin of 𝜃, for every instance of 𝑦 in this equation. Using this substitution gives us that 𝑟 sin 𝜃 is equal to two 𝑟 cos of 𝜃 plus three. And this is an equivalent equation to the equation we got in the first part, except now we’ve eliminated the variable 𝑦.
Finally, the last part of this question wants us to make 𝑟 the subject of our equation. This means we want to rewrite the equation we got in the second part of this question to be of the form 𝑟 is equal to some function of 𝜃. We’ll start with the equation 𝑟 sin 𝜃 is equal to two 𝑟 cos 𝜃 plus three. Next, we’ll subtract two 𝑟 cos of 𝜃 from both sides of our equation, so that every term with an 𝑟 is on the left-hand side of our equation. This gives us that 𝑟 sin 𝜃 minus two 𝑟 cos 𝜃 is equal to three.
Next, we notice that both terms on the left-hand side of our equation share a factor of 𝑟. So let’s factor this out. Taking out the shared factor of 𝑟 from both terms in our equation gives us that 𝑟 multiplied by the sin of 𝜃 minus two cos of 𝜃 is equal to three. Finally, we can make 𝑟 the subject of this equation by dividing both sides of our equation by sin 𝜃 minus two cos 𝜃. Doing. This gives us that 𝑟 is equal to three divided by the sin of 𝜃 minus two cos of 𝜃. Therefore, what we have shown is that Cartesian equation, 𝑦 is equal to two 𝑥 plus three, is equivalent to the polar equation, 𝑟 is equal to three divided by the sin of 𝜃 minus two cos of 𝜃.
