I’m still trying to deal with the backlog, and the other “topical” posts are requiring a bit more work than envisaged. So, to keep things ticking along, and by request, here is a new TNDOT. As for TNDOT 2, this one is from the 1913 Victorian Matric Trigonometry Honours Exam. Have fun.
![\color{blue}\boldsymbol{\aligned &\mbox{\bf {7. From the equations}}\\[3\jot] &x^2 + y^2 = \frac{x\cos 3\theta + y\sin 3\theta}{\cos^3\theta}=\frac{y\cos 3\theta - x\sin 3\theta}{\sin^3\theta}\,.\\[3\jot]&\mbox{\bf \quad \bf{(i) Eliminate }} \theta \mbox{ \bf{and express the result in a simple form; }} \\[3\jot] &\mbox{\bf \quad \bf{(ii) Find }} x \mbox{\bf { and }} y \mbox{\bf { in terms of }}\theta. \endaligned}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-92a9be13c0e55091ff9cfcc0cd15b703_l3.svg "Rendered by QuickLaTeX.com")
UPDATE (05/04/24)
OK, after a very long delay, here is a tidy solution to the question. I’ve followed the hints of J. D. and the notes of Red Five. (I also suspect, as do Simon and Red Five, that there may be some neat complex numbers approach to the problem, but I didn’t pursue it.)
To simplify the writing a little, we’ll first set
Then the two equations we are given can be written as
![\color{Plum}\boldsymbol{\aligned &(a)\quad &&\phantom{ioj}cx + sy = r^2\cos^3\!\theta\,, \\[3\jot] &(b)\quad &&-sx+cy = r^2 \sin^3!\theta\,.\endaligned}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-294262a88b05a569c7274b0f66ad2dc0_l3.svg "Rendered by QuickLaTeX.com")
Thinking of this as two linear in equations in x and y, this system can then be easily “solved”, either by matrix trickery or, as was probably expected in 1913, by combining the equations trig-naturally (c(a) – s(b) and s(a) + c(b)), and using 
:
![\color{Plum}\boldsymbol{\aligned &(c)\quad x = r^2\left(c\cos^3\!\theta-s\sin^3\!\theta)\,, \\[3\jot] &(d)\quad y = r^2\left(s\cos^3\!\theta+c\sin^3\!\theta)\,.\endaligned}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-d63eab12268d876b06524f719a264cdc_l3.svg "Rendered by QuickLaTeX.com")
The next step is to write the triple-angle guys and the cubic guys as the same kind of thing. It seems easiest to convert the cubic guys, using the following formulas (which would have been very well known in 1913):
Plugging these into (c) and (d) and simplifying, we get,
Now we’re home. Squaring (e) -1 and (f), and adding, we get
Multiplying through by 
, expanding and using 
, this easily simplifies to
which is the desired answer to (i).
Next, to write x in terms of 
, we can multiply (e) through by 
and plug in (g):
This is a linear equation in x, which unravels to give
Finally, to write y in terms of we can plug (g) and then this expression for x into (f), giving,
Getting a common denominator, this easily simplifies to
and we’re done.
I love this question. My solution runs into a few pages though, so I’m very curious to see what others come up with…
As with others, use the sum-to-product identities at one point (well, I did).
Other hints on request.
There’s an interesting 90 degree rotation in theta hiding in there. Maybe could be exploited for a nice solution, but I haven’t had time to play yet!
A very interesting observation. I missed that (my method is more brute-force than finesse).
After eliminating
, finding an expression for 
was easy enough. The expression for 
took a few more lines of scribble though, so again, I’m fairly certain I’ve missed the “nice” solution.
Would writing
help at all I’m now wondering…
Maybe.
Thanks for the idea.
UPDATE: While the rotation idea is certainly a nice one, I cannot (yet) use it to simplify my solution at all. Complex numbers don’t seem to help either (I’m sure a solution using the complex exponential forms of sin and cos would work, I’m just not convinced it is an improvement on the solution using product-to-sum identities).
If no one has posted further ideas by Monday, I might drop a hint or two as to how I went about it. Full solution (remembering that I really don’t think it is the best solution possible) available on request.
Thanks, RF. I’ve only glanced at it. I have some thoughts, but haven’t tried them out.
If you temporarily consider the
on the left to be a constant 
, then the system becomes a 
linear system in the unknowns 
and 
. The determinant of the system being 1, it’s not hard to write down the solution using Cramer’s Formula, and after that the expressions for 
and 
can in fact be simplified considerably (without ever changing 
back to 
). Once that’s done, 
can be eliminated by using 
. In principle, when 
is eliminated, there should be an 
factor in the equation obtained, because 
is always a solution.
Cramer’s rule. That is genius. Thanks. I’ll have a go to see where that leads.
I did think of a geometric approach, but didn’t get very far.
UPDATE: I think (still checking my work) that this leads to
From there
and 
should be not too difficult to extract.
Yes, I also have that result. After simplifying the expressions for
and 
, I get 
. Eliminating 
leads to 
. After that, on the assumption that 
, we can substitute 
for 
in the first equation to obtain an expression for 
in terms of 
, and then again put 
into the second equation to find 
.
Very nice, J. D. (Just calling it “the inverse of a 2 x 2” is less scary than “Cramer”.)
I doubt students at the time (and much less now) would have been expected to have approached the question in such a manner. But probably they would have done the simultaneous equations work that mimicked your approach.
Your comment made me wonder whether Cramer’s Rule was taught at the School Certificate and Higher School Certificate levels in England in the past. It seems determinants are not mentioned in Barnard and Child’s “New Algebra” (1912), but Cramer’s Rule is discussed in Ferrar’s “Higher Algebra for Schools” (1944), which was for mathematics and science specialists at HSC level. The 2 x 2 version is discussed in Part III of Durell’s “New Algebra for Schools” (1931), which was for “Additional Mathematics” at SC level.
Thanks, J.D. I think of matrices and determinants as a more modern thing, particularly as a school topic. I haven’t hunted, but I don’t think these were school topics in Australia at the time of this question.
Matrix determinants were a thing in some of the 1900s exam papers up to 3×3.
Matrix operations beyond determinants do not seem to have been a thing.
Thanks, RF.
That is the Victorian papers. I have yet to get my hands on the NSW papers of the time.
If anyone has any though… I am very interested. The NSW state library apparently has paper copies (bound into annual volumes) back to pre-1900 but won’t allow inter-library borrowing.
Here is my solution. It is not pretty and I’m convinced there must be short-cuts I’ve missed.
1913 Trigonometry Honours Q7
That’s some good work — your solutions for
and 
do solve the problem… but I agree that it seems like too much for an exam question and there must be some other way through it all. I played a bit with this question, but struggled to find a simple way to eliminate θ or solve for x and y. Maybe you path through isn’t so bad… especially now that I review it again, it is fairly logical.
I’m still in NPLAN mode, but once I’m free I’ll try to de-matrix J. D.’s method.
Thanks Simon. I struggled with this question for DAYS, convinced that I was missing something before I decided to just brute-force it.
I still think there may be a complex numbers solution hiding somewhere (as a few papers around the same time made passing reference to De Moivre’s theorem and Euler’s formula).
OK, I finally got around to updating the post with solutions. Thanks to everyone for their contributions. I’ll try to update TNDOT 3 and to post a TNDOT 5 in the next couple days.