Polar graphs? Please help!?

Question

I have to draw a polar graph (i.e. a flower) and I'm seriously struggling with this. I also need to locate 5 points on it and then create a table for them (radians & fractions)....can anyone please help??

Answer 1

A flower? So I'm guessing you're dealing with an equation of the type:

r(t) = sin nt or r(t) = cos nt. Is that right?

The most important step in drawing a polar graph is to first draw the graph of the r(t) in cartesian coordinates. That is, draw y = sin 3t ( or whatever your function is) with t as your x-axis and y as the vertical axis.

So you should get a sinusoidal wave. Now this tells you how to draw your polar graph - it tells you how far out the point should be at each angle. So, to make it simple, try drawing each point at multiples of 30 degrees (pi/6). Suppose you have r(t) = sin 3t. Start at 0 degrees. sin 0 = 0, so the first point is at the origin. Then at 30 degrees from your x-axis, your point should be sin 3(30) = 1 unit away from the origin. Then at 60 degrees, sin 3(60) = sin (180) = 0. Therefore, at 60 degrees, the point is back at the origin. So, the first petal of the flower starts at the origin, reaches its farthest point at 30 degrees, then returns to the origin at 0. See?

The tricky part is that when you obtain negative values for r(t), you have to take its absolute value and flip it by 180 degrees. For example, at 90 degrees, sin 3(90) = sin 270 = -1. Therefore, you draw at point 1 unit from the origin. Except, you have to draw it not at 90 degrees, but (90 + 180 degrees), which is 270 degrees. Take a look at the polar graph of sin 3t: http://www.ies.co.jp/math/java/calc/sg_kyok/sg_kyok.html

It's simple to create a table with point values. Choose any 5 angles. You might as well choose the same 5 of the same angles you used to graph your polar graph, i.e. 30, 60, 90, 120, 150. Plug them into the function r(t) = sin 3t, where t is the angle theta. Pi radians is the same as 180 degrees. So 30 degrees is pi/6 radians. 60 degrees is pi/3 radians. 120 degrees is 2pi/3. And so on.

Hope that helps.

Answer 2

r = cos 2x Is the graph of a four-leafed rose.

Using the properties of the cosine , the tests for symmetry show that the graph is symmetric with respect to the polar axis and its extension, the 90 degree axis and its extension and to the pole.

Set r = 0 getting cos 2x = 0 or 2x = pi/2,3pi/2, etc. So x = pi/4. 3i/4 etc. These angles give the direction of the tangents to the curve at the pole. As x varies from 0 to pi/4, r varies from 1 to 0 giving a a half loop in the 1st quadrant. Since the graph is symmetric to the polar axis, the reflection of this half loop across the polar axis completes the loop and gives one of the four petals of the rose.

Now as x varies from pi/4 to to pi/2, r varies from 0 to -1 giving another half loop in quadrant 3. Since the graph is symmetric with respect to the extension of the 90 degree axis, the reflection across the 90 degree axis completes the loop and gives us a 2nd petal of the rose.

Now since we have symmetry with repsect to the pole, we can now draw the other two petals and have our required graph.

You can get as many points as you like by picking a value for x and solving for r.

Note: I used x instead of theta for the angle.

Answer 3

go to wikipedia and type in polar graphs.You will be taken a page that will show you the graph and complete descriptions. I can explain it to you but would take forever..LOL.. good luck

Answer 4

Try using
r = cos(4πt) as 0 ≤ t ≤ 1
That'll give you a nice, four petal flower.

Doug