Trigonometry: Part 1 (2 Unit)

[jamonwindeyer]

Yes everyone, the time has arrived. You stared at posters blankly when you first got to high school, and now you sit an exam where about a quarter of your mark will likely depend on understanding what many view to be the worst part of the HSC Math Courses... Trigonometry. To them I say, read this guide! My aim is, by the time you finish reading this, you will be a master of the triangle. In recent years, BOSTES has cut back on the amount of trig in their exams, but it is still a heavy contributor, and this could be the year they go all out like they did in a few papers in the 1990's.

This guide is in two halves. This half will cover the 2 unit part of the course only; for extension students, the second half is either already in your forums or will be very shortly. Once again, I'll insist that if absolutely anything at all is even the slightest bit unclear (I have to sacrifice some explanation for keeping these short), let me know below! You can register and also take advantage of the awesome notes available for HSC Math... Who needs study notes when you have websites like this!

Okay, 2 unit trig. We'll start simple; obviously, you should know the three trigonometric ratios; sine, cosine and tangent (SOHCAHTOA for the win!). It is highly unlikely you'll get asked a simple opposite over adjacent style question, but you need this as the building blocks. You should also know the reciprocal functions (don't do what I always did and confuse secant and cosecant):





You should know the graphs for these functions, and YES, they are functions as well! To save space, they are all on the same graph. Sine is in blue, cosine in red, tangent in green. Remember that the tangent function has a vertical asymptotes all along it's domain (at odd multiples of \(\frac{\pi}{2}\)). The sine function (and by extension, the cosine function) can be determined by the general formula:

A is the amplitude of the function, it oscillates between A and -A (if a constant is added, this shifts the whole graph up or down). n determines the period of the function: If n is doubled, the graphs period is halved, etc. Making n bigger "squeezes" the graph closer together. \(\Phi\) shifts the graph left or right, the easiest way to check is a substitution, rather than remembering a rule.

And finally, before heading into the harder stuff, remember the following relationships apply for acute angles:



I'll mention again, questions on this content alone are uncommon. There is too much else to examine for trig. However, it could be the trick to a calculus/geometry proof, or the step in a later question, so don't forget the geometrical basis of these ratios. It often gets lost in all the rules and equations we are about to look at.

The fun begins when we start looking at angles of any magnitude. Lots to remember here, but we can condense it to a few key points, and a diagram which was on my wall for about 3 months leading up to the HSC!

• It is convention to measure angles from the positive x axis, taking counter-clockwise as positive
• An examination of the behaviour of the trig functions reveals they oscillate with a period of \(2\pi\) radians/360 degrees. Thus, the properties of any angle can be determined by adding/subtracting 360 degrees until we get into the appropriate range
• Further, the properties of every ratio repeat in some manner from the first quadrant (acute angles). Therefore, we need only know these, and a method of extrapolating for larger angles. This method is the method you will recognise from the figure below:

The rhyme scheme is obviously optional, I live near Sydney, so all stations to Central is currently my mantra. But this diagram is awesome to remember how the angles are expressed and what their signs are!

This idea is rarely asked by itself either, but it can happen for a couple of easy marks in an algebra proof. I'll demonstrate the thinking with an evaluate question, even though it could obviously just be solved with a calculator (though for exact form, this isn't always the case!). Nice and easy.

Example One: Evaluate \(\cot{-135}\)



Normally, we use these angles of any magnitude to solve trigonometric equations in given ranges. The simplest of these require no simplification.

Example Two (HSC 2007): Solve \(\sqrt{3}\cos{x}=\sin{x}\) on the interval \(0\le x\le 360\)



Other times, they will need to be simplified. This relies on the pythagorean identities. The first of these can be used to easily derive the other two.



These can be used to solve, among other things, quadratic trigonometric equations.

Example Three: Solve \(2\cos^2{x}-3\sin{x}=0\) for \(0\le x\le 360\)



As I mentioned previously, BOSTES is shying away from asking these sorts of questions flat out. Instead, they will ask you to find where two trig functions intersect, to set up an integration question. Or use trig functions in an acceleration/velocity question. Of of course, in geometry. This also suggests you should know your trig derivatives and integrals. I found it easier to just remember the derivatives, the integrals are easy to work out from these and are provided on your reference sheet anyway (just watch your signs):



The final part of trigonometry in 2 unit is the sine and cosine rules (as well as the sine area formula). I also mention and do an example for this in the geometry guide. Let's do that whole question here, the second part uses the sine rule. The rules, if you forget:



Example 2: Chris leaves island A in a boat and sails 142 km on a bearing of 078° to island B. Chris then sails on a bearing of 191° for 220 km to island C, as shown in the diagram.



a) Show that the distance from island C to island A is approximately 210 km.

This is a good chance to brush up on bearings also 



Therefore, by the cosine rule, the required distance d is:



b) Chris wants to sail from island C directly to island A. On what bearing should Chris sail? Give your answer correct to the nearest degree.

We now have the distance AC from the previous part. So we apply the sine rule:



Then we apply some basic bearings-typical geometry to convert this to our bearing. We note that angle BCN is 11 degrees by co interior angles to help with this proof.



This is to the nearest degree, like all your bearings should be.

And that is all you need to know for trig! How about that for efficiency, you have a guide with formulas and solid examples, readable in 10 minutes before an exam!

Of course with having to cover lots of stuff, I may occasionally not give enough detail to something you really need help with. Don't despair! Just ask a question below or in the question thread . I do my rounds often to answer any questions, and there are lots of other knowledgeable people ready to help as well.

Question Thread Link: 2U Maths Question Thread: Ask Your Questions Here!

Stay tuned for more guides guys, and remember to register to get access to an AWESOME set of notes for both 2U and 3U, get the best set of study notes in town, for free!

 

A GUIDE BY JAMON WINDEYER