On our maths course there are two broad types of proofs that we have to be able to do in trigenometry( there are some other more niggly bits but we can cover them later).
The following two example typify them. Students have trouble with these because they simply dont have a logic or method by which to start or “get at it”- in these notes I hope to give that method concisely.Take out a notebook and start taking down the important points of what I say here and follow the maths with me(dont just read it), its not as long as it looks.
1. sec2A = 1 + tan2A
2. a(bcosC – ccosB) = b2 – c2
type number 1: you have to change what you see(the thing to be proven) into an undeniable truth.
1. Simplify any fractions in the expression(starting with the side that is most complicated) in this case there are none but otherwise you would have to find a common denominator.
2. Change everything to sines and cosines, keep in mind that :
- sec = 1/cos, cosec =1/sin and cotan =1/tan = cos/sin.
lets do that with our example: sec2A = 1 + tan2A
sec2A = (1/cos)2 = I/cos2 and tan2A = sin2/cos2
So now it looks like this : I/cos2 = 1 + sin2/cos2
3. Naturally you want to have a tidier expression and since you see cos2 on both sides, multiply across to cancel them on the bottom. Keep in mind, when you can cancel neatly and slice those messy fractions off the page-DO IT!
4. Look out for cos2 +sin2 = 1 or variations of it like 1 -sin2 =cos2 or cos2 = 1 – sin2. Aim towards these and when you see them-make the suitable change.
So we get : 1 = cos2 + sin2 after we multiply across by cos2
which is undeniably true. so basically we changed the thing at the start into something that was true. so the thing at the start must have been true-you proved it-well done!
Now take out a page, dont be afraid, and practice the ones in your book. post any questions here and try the above guidlines. The Key is really point 2 and 4.
type number 2: really we use subsitution.You know that it is type two when you see the names of sides of triangles (b, a ,c).Refer to labelled diagram of the names of triangles. type one will only have angles big letters (A,B,C)
image from campus northpark
- Sin A/a = Sin B/b => sinA = asinB/b and sinB =bsinA/a….SINE RULE
- a2 = b2 + c2 -2bccosA => cosA = (b2 + c2 – a2)/2bc …. COSINE RULE. Write this one out for cosB and cos C yourself.
- if you see cos in the thing you have to prove use the second one, if you see sides and no cos mentioned use the sine rule(first one)
- Now what we do is (substitute) put in the expression for Cos A,B,C or Sin A, B,C into what we are asked to prove.look below at the example
learn the above off by heart, you dont want to work them out on the day.
our example: a(bCosC – cCosB) = b2 – c2 = abCosC -acCosB =b2-c2(remove brakets)
no sines, sides of triangles mentioned, cos: we must use the cos rule. see the Cos C, Cos B? well we have to put in the rearreanged expression in blue for them.
Cos C = (a2 + b2 -c2)/2ab and Cos B = (a2 + c2 – b2)/2ac
so the example becomes ab(a2 + b2 -c2)/2ab) – ac (a2 + c2 – b2)/2ac
which boils down to (a2 + b2 -c2)/2 – a2/2 -c2/2 +b2/2 = b2 – c2. thats it!
BY AXLE( Ahmad Asad)