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In this video, we're going to go through the answer to question number 37 from chapter 11 .4.
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So it has to plot the functions r equals cause of m theta for m equals a third, two, three, and seven.
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Firstly, for m equals a third.
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So one thing to note about all of these curves is that because it's r equals cause of something, we know that cos is an even function.
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So cause of minus theta is equal to cos of theta.
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So therefore we have x -axis symmetry in all of these curves.
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So starting with m equals a third, let's think about some easy to calculate values and put them in a table first.
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So theta and arc.
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When theta is equal to zero, r is equal to one.
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When theta is equal to pi by 2, r is equal to cos of pi by 6, which is equal to root 3 over 2, which is around about 0 .9.
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When theta is equal to pi, r is equal to, well, it's cos of pi by 3, which is half.
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And when theta is equal to 3 pi by 2, then r is equal to cause of pi by 2, which is just 0.
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And when theta is equal to 2 pi, r is equal to cos of 2 pi by 3, which is minus a half.
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Which takes us back to that.
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So we can keep going with more values of theta, but we know that we have x -axis symmetry, so i think we've got all we need right now.
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So starting from here, we just join the points up in order.
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And then that's where we've got two.
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But now we can just reflect that in the x -axis, and that is the curve for r equals cos of theta over three.
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Next up is equal to two.
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Again, we're going to have x -axis symmetry, but let's draw some axes and calculate some known values.
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When theta is equal to zero, r is equal to 1.
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Let's try when theta is equal to pi by 8.
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Then we've got r equals cos of pi by 4, which is root 2, over 2, which is around about 0 .7.
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So that's r equals 1 here, pi by 8, then r is equal to 0 .7.
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Then r is going to be equal to cos of pi by 2 which is 0, 3 pi by 8, which is along here, then cost is going to be equal to, sorry, r is going to be equal to minus root 2 over 2.
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So that brings us down here.
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What about pi over 2? well, in that case, r is going to be equal to cos of pi, which is minus 1.
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That takes us down here.
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Then what about 5 pi by 8? that's going to be minus root 2 over 2 again.
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So 5 pi by 8 is this one.
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So minus takes us down here, minus root 2 over 2.
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What about 3 pi by 4? that's going to be 0.
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And then 7 pi by 8, that's going to be root 2 over 2, which takes us there...