>> All right another aspect that can be evaluated before graphing and sometimes this is helpful if for no other reason than to set up your graph paper, is to figure out the maximum R values. Now to figure maximum R values all you do is go through and analyze the greatest and least value that the Trig function can take on. Now cosine theta can take on values, you see, what, for different angles, what values do we get to the cosine of theta, well it varies between negative 1 and 1. So let's take the extremes. If cosine of theta is negative 1, you see, negative 1 times negative 2, oh that would be 2. 1 plus 2, oh that's 3. Okay now what about the other extreme for the cosine of theta could be 1. If it's 1, we have negative 2 times 1 that's negative 2, 1 minus 2 that's negative 1. What's the largest value? You see, now in terms of distance, by the way, that negative 1 that I found is still a distance of 1 from the pole isn't it? Now it's just taken in a negative direction, you see, from some angle, but distance from the pole, it's kind of like an absolute value, you see, you think of it as 1. So the maximum value though was found when cosine theta takes on the value negative 1 in which case we have an R value of 3. So the maximum R then would be a distance of 3 from the pole. For 5 cosine 4 pi, well let's see, now don't let the 4 pi idea scare you, it's just 4 times some angle, it's just a bunch of angles that we have over here that we're taking the cosine of. But the cosine of angles, no matter how we describe them, change from negative 1 to 1. You see? And so let's see, if this value turns out to be negative 1, 5 times negative 1, negative 5. Distance from the pole, 5. If this angle, see, for its maximum value, or the greatest value that this could take on would be 1. 1 times 5, distance of 5. So the maximum R that this can take on would be 5. Okay. Let's consider a probable. Let's go back to the graphing calculator now but before we do let's analyze a couple things. We have R equals 2 plus 4 sine theta. Now because our expression here contains only sine, the sine, now you can look at the SIGN pattern for the sine curve or in what quadrants is the sine positive and negative to discover aspects of the symmetry idea. But it's the notion that when sin is involved, S I N, Sin is involved, the sine function is involved that we have symmetry about the theta equals pi over 2 line. So here we're going to have, this is symmetric to the pi over 2 line, the maximum R value, well let's have a look, sine theta can take on values from negative 1 to 1 if the value if negative 1, negative 1 times 4 is negative 4, 2 with negative 4 is negative 2, so the distance there is 2. Let's keep that in mind, all right, now sine can take on the value 1, 4 times 1, 4, 4 plus 2 is 6. Okay 6 or 2 oh it's going to be 6. So the maximum R is 6. Now when does that occur? Well we can tell when that occurs. Now sine is, let's see, that 1 sine of the angle occurs when the angle is how big? Well let's think about the sine curve, it starts out at the intercept and it goes to the top of the amplitude. That's where it's 1. Okay top of the amplitude occurs at pi over 2. So at pi over 2 that's going to occur. Okay. So we get this max when theta is pi over 2.