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There is a proof using sums to products at http://en.wikipedia.org/wiki/Law_of_tangents

Suppose you have \triangle ABC with sides opposite angles A, B, C named a, b, c. Then {a-b\over a+b}={\tan({1\over2}(A-B))\over\tan({1\over2}(A+B))}

Compare Fitzpatrick's 3 unit lovely proof (page 82) of the derivative of sin x to Jones & Couchman's horrible one (Book 1, page 440). Fitzpatrick uses a sum to product and gets it out in just a few lines. Jones & Couchman's is a page of ugly algebra nobody really wants to read. Another...

Yes indeed. Products to sums and sums to products is VERY useful and greatly simplifies many things. And this is why it should still be taught. The fundamental limit is \lim\limts_{x\rightarrow0}{\sin x\over x}=1 (provided x is in radians)

Yes there were some other changes. Another example is that division of an interval in ratios was originally in 2 unit. But is now 3 unit. Others like this are angle of intersection of lines, composite angles in trigonometry, fundamental limit. Other things were completely removed, like...

The 2 and 3 unit courses (previously named Level 2S and Level 2F - S for short, F for full) didn't change as much. There used to be 3D coordinate geometry and Hooke's law which are no longer in the current courses. The 4 unit course changed more because originally it was a 2 year course...

By the way, you can get the older papers (1967-1994) at http://www.angelfire.com/ab7/fourunit/2u1967-1994.zip and newer ones (1995-2009) at http://www.boardofstudies.nsw.edu.au/hsc_exams

Yes. You are right. It was in the 1988 2 Unit Q10b (but with plusses, not minuses, and real roots):

Correct! This is in my longer pdf file: http://users.tpg.com.au/nanahcub/1916-1989-2001.pdf

Solution: http://users.tpg.com.au/nanahcub/1989.pdf Next Question: Do you know what Coroneos said about it in 1989?

I made a website for Mathematics Extension 2 which might be of some use: http://users.tpg.com.au/nanahcub/me2.html

Yeah. rly. My solution is attached.

Actually, his final equation is incorrect. The correct equation is {1\over2}L\sqrt{\pi(4-\pi L^2)}-({\pi L^2\over2}-1)\cos^{-1}({\pi L^2\over2}-1)={\pi\over2} And when I put that one into wolframalpha, I get L=0.653742534637021.... Subsequent check of the sum of the areas of the 2 segments...

The solution to jetblack2007's final equation is L=\sqrt{2/\pi}

This is a 2 mark question. P(1)=1-(n+1)+n=0 (This gets you the first mark) P'(x)=(n+1)(xⁿ-1) &there4; P'(1)=0 (This gets you the second mark) &there4; x=1 is a double zero. For 1 < k < n+2, P^(k)(x)=(n+1)n(n-1)...(n-k+2)x^n-k+1 &there4...

Sedenions cannot be represented by matrices because they are not associative. The real projective line and Riemann sphere are one-point compactifications of the real numbers and complex numbers. They are not generalisations to higher dimensions. If you think number systems more general than...

Didn't your teacher teach you that you can't divide by 0? Suppose we ignore the teacher and decide we are going to use sedenions. Then we can divide by 0. Every sedenion constructed with the Cayley–Dickson construction is a real linear combination of the unit sedenions 1, e₁...

Are you comfortable with zero divisors and are happy with sedenions, etc as numbers? Most people are not. We could start with natural numbers, then integers, rationals, reals, complex numbers, quaternions and LASTLY the octonions. Furthemore, we should ask what these numbers are useful for...

It presents a problem for extending number systems beyond the octonions (8 dimensional). For example the sedenions (16 dimensional) cannot be a division algebra. Two nonzero elements can be multiplied to give 0. All higher-dimensional hypercomplex numbers likewise contain zero divisors. So we...

A few days ago, njwildberger posted the following on youtube: We think of natural numbers. We think of defining objects called fractions, integers, rational numbers. So we think of the concept of ''number'' in quotes as something that's open, that keeps on going. There's no bound. So we for...