Match each equation with its polar graph from choices A–D.
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Question

Match each equation with its polar graph from choices A–D.

r = 2/(cosθ + sinθ)

Accepted Answer

Welcome back, everyone. Graph the given polar equation R equals 5 divided by we call sine theta minus sine theta. What we're going to do is basically introduce a coordinate plane, and essentially we are going to calculate the R values for different values of data. So first of all, if our theta is 0, then the value of R at 0.0, which means that we're substituting cosine at point theta, gives us cosine of 0, and sine theta gives us, gives a sign of 0, right? So we're substituting 0 for every theta. This gives us the R value of 5/3. Which is approximately equal to 1.67. So what we can do is just add a point that corresponds to approximately 1.7 on the given graph, right? So essentially on the given graph we want to add a point that is approximately 1.7. Now what we're going to do is evaluate R at I divided by 6. Now let's label pi divided by 6. In the given graph, pi divided by 6 essentially corresponds to 30 degrees, right? This is our angle pi divided by 6, and then we can evaluate the function. We get approximately 2.38, and we can keep evaluating these values at perfect angles, right? Then we can move on to pi divided by 3. So let's choose by divided by 3. R and pi divided by 3. Gives us a value of about. 7.9. Which is done for by R at pi divided by 2, right? Let's label pi divided by 2. Now this gives us an exact value of -5. Then we can evaluate R at 2 divided by 3. Which gives us about -2.11. So let's label 2 pi divided by 3. Then let's evaluate R at 5 by divided by 6. This gives us approximately -1.61. Now, let's label 5 by divided by 6. Followed by i, so we are going to evaluate R at pi, which gives us about. -1.7. Then we are going to evaluate R at 75 divided by 6. Which gives us about -2.38. So let's label that angle 7 pi divided by 6. Now let's evaluate R at 4 pi divided by 3. Which gives us approximately -7.9. And then we can estimate R if we divided by 2, which gives us exactly 5. So let's add our points. We divided by 2, or pi divided by 3. And now what we're going to do is basically add these points. So let's begin with the first one. R of 0 gives us about 1.7. Then we have R at pi divided by 6. This is about 2.4. So we want to have approximately 2.4. Then we're going to add. About 8 at pi divided by 3. Then we have -5 at 5 divided by 2. So this means that we're going down and we have -5. Followed by -2.11 and 2 divided by 3. So since it is a negative value, we're going to look at the opposite quadrant, and we want to add about -2.11. So let's remove the line and we're going to add about -2.11. And now what we can notice is that all of these points, they're actually in a straight line. If we move on, we will see that all of these points, they make a straight line. So what we can do to plot it accurately is basically take two points, right? So at 3 by divided by 2. We have 5. And now let's consider the other point. We also had pi divided by 3. Right, this point gave us about 7.9. So essentially we can connect these two points. And we will notice that all of the other points, they lie on the same line. And that's it we have. The graph of the given polar equation. Thank you for watching.

Match each equation with its polar graph from choices A–D.
r = 2/(cosθ + sinθ)

Key Concepts

Polar Coordinates and Graphing

Conversion Between Polar and Cartesian Coordinates

Trigonometric Identities and Manipulation

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