Question

Match each polar equation with its description. An answer may be used more than once or not at all.

Answer 1

1. h

2. b

3. e

4. g

5. h

6. c

7. a

8. g

9.b

10. d

1) There are some families of Polar Equations that we can identify by the trig function, the number of terms.

2) So examining that table we can state that

1. r=8-8cos(θ)

Note that this can be written from this pattern:

`r=a\pm b\cos (\theta),a>0,b>0`

So this is a Limaçon, but since a= 8 and b=8 we can see a special case, for Limaçons: a Cardiod

2. r=5sin(4θ)

In this equation, we have a Rose Curve for the value of the coefficient of that sinusoidal function is a constant than the value of the angle.

3. r=9+4cos(θ)

We have here a similar case to the first one, note that in this case, the quantity of "a" is greater than the coefficient "b".

So this is a Dimpled Limaçon

4. r=3

If we convert this to a rectangular form, we'll clearly see the equation of the circle.

`\begin{gathered} x^2+y^2=3^2 \\ x^2+y^2=9 \end{gathered}`

So this is a Circle

5. r=5-3sin(θ)

This is clearly a Limaçon, but this time a> b, so that will make a change in the Limaçon, again we can see a Cardiod

6. θ=2π

This is another case, which we can see better when we convert it to the Rectangular coordinates:

Plotting that we can see a line.

`\begin{gathered} \tan (\theta)=2\pi \\ (\theta)=\tan ^(-1)(2\pi)_{} \end{gathered}`

7. r²=25sin(2θ)

Note that generally, we can write this one as r²=a²sin(2θ) and this is the formula for a Lemniscate

8. r=6sin(θ)

This equation when plotted yields a circle, for whenever we plot r=asin(θ) or r=bcos(θ) we end up tracing Circles

9. r=-3cos(5θ)

Similarly, to the 2nd item, this produces a Rose Curve note that this follows the same pattern.

10. r=2+6sin(θ)

This is also a Limaçon, as in some previous cases but since the coefficient a=2 and b=6 and a Limaçon with a Loop

2) Thus the answer is:

1. h

2. b

3. e

4. g

5. h

6. c

7. a

8. g

9.b

10. d