Well, I have a test tomorrow, and I've actually slacked off for the past week or so, so now it's time to do some power studying. (Actually I've already done some power studying, now this is just me writing up about it.)
My idea of power studying is going through all of the topics listed in a book or course outline and writing down stuff on each of the dot points or book headings. If there's something that I can't write anything for, then I revise it.
All right. Here we go... a speedy review of chapters 1-6!
Chapter 1: Polar Coordinates
Distance between two points
Just use cosine rule (i.e. c^2 = a^2 + b^2 - 2ab cos (theta)). Polar coordinates give the lengths of two sides of the triangle so a and b are all worked out for you. As for cos (theta), theta is simply the difference between the angles given in the polar coordinates. Then solve to find c. If you get stuck, just draw a diagram and it should all make sense (if it doesn't, then revise the cosine rule and/or polar coordinate basics).
Polar equations and graphs
If you have theta = (angle), then the graph is basically a line in the direction of the angle. (i.e. if the equation is theta = pi/4, then you have a line continuing at pi/4 radians, or 45 degrees, as measured anticlockwise from the positive x-axis). If it says anywhere that r doesn't have to be greater than theta, then the line extends in both directions.
If you have r = (constant), then the graph is a circle with a radius equal to the constant.
You can also get inequalities for these as well. Remember, if it's a greater than/ less than sign without the equals bit underneath, you need to draw a dotted line, not a solid line.
Oh yeah, and there are the spirals too, in the form r = k(theta), where k is a constant, r is the magnitude and theta is the angle. Normally you see these in those questions where it asks you to write the equation for the graph. Normally it helps to use the values of r at (pi/2) and/or pi to help you determine k, and, therefore, the equation.
Chapter 2: Complex Numbers
Polar form of a complex number
Okay, there's an explanation for this, and while I tend to value understanding over memorisation, this brief summary is an exception. Basically, the polar form is in the form r cis (theta) where r is the modulus (i.e. the length between the origin and the location of the number on an Argand diagram), cis is short for "cos theta plus i sin theta" (ties in with the explanation that I'm missing here) and (theta) is the argument, or the angle between the positive x-axis and the line between the origin and the location of the complex number. Normally, theta is stated in radians, and the principal argument (the one normally used) is between negative pi and pi (including positive pi, but not including negative pi).
By the way, that old familiar form of a complex number a + bi is also called the Cartesian or rectangular form of a complex number.
To convert from rectangular form to polar form, first find the modulus by using Pythagoras' theorem (i.e. modulus = sqrt(a^2 + b^2)). Then use SOHCAHTOA to find the angle/argument. Sometimes it helps to draw a diagram to make sure you get the right angle. Remember, the angle you want is the one measured from the positive x-axis.
To convert back, draw a diagram and use SOHCAHTOA or just have a couple of formulas memorised. The ones you'll want are x = r cos (theta) and y = r sin (theta). (Hopefully I've got them right there...)
Multiplying and dividing complex numbers in polar form
Believe it or not, there ARE benefits to polar form- it's much easier to multiply and divide in this form!
To multiply: multiply the moduli, add the arguments.
To divide: divide the moduli, subtract the arguments.
Then just make sure that the argument fits into the definition of a principal argument (between negative pi and positive pi- but not including negative pi). Done!
Complex numbers- regions on a diagram
For any complex number z, if you ever get the equation |z| = (constant), then the diagram is basically a circle with the centre at the origin and a radius equal to the constant. Apparently, there's a fancy word for it- the "locus" of |z| = (constant).
Then, if you have something like |z-w| = (constant), where w is another complex number, then the diagram is basically a circle with its centre at point w and a radius equal to the constant.
And then you can do all kinds of inequality stuff here too.
Okay, stuff what I said about chapters 1 to 6. It's 9.27pm, and I want to get to bed early-ish because I have orchestra tomorrow and I think that sleep will help me more than studying. So good night, everyone :)
(By the way, I found out that I got 30 page views on the 31st of March. I was kind of surprised, to be honest, because I didn't think that anyone but me ever came on here since my blog is so nerdy, and even then I haven't updated my blog in a few months. Hmm. I'm curious now.)
And now you have a comment to prove someone does come here.
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