(Yes, there will be a Secret General Exam Business post, later today, and good luck to you or your students.)
Ok, no one had too much to say about last week’s TNDOT, which is fine. Although there were a couple of things to note, I think, it wasn’t of that much interest, and was posted mainly to complete that particular exam (Pass Algebra 1911). I’ll update the TNDOT soon.
This one is also part of completing an exam, but it should be more challenging and more interesting. The questions are from the 1913 Matric Trigonometry Exam, of which Q2 (TNDOT 2) and Q7 (TNDOT 4) have already been posted. (See also the other trig posts: TNDOT 1 and TNDOT 8.)
The exam consisted of nine questions in total, and here are the first three (repeating Q2 from TNDOT 2). Have fun.
1. (i) If 
then each is equal to
and to
(ii) Between what limits must 
lie, for
and 
to give real values of 
and 
.
2. Sum
![\color{blue}\boldsymbol{\aligned &\mbox{\bf \quad {(i) }} \sin(k+1)a\,\cos a \, + \, \sin(k+2)a\,\cos 2a \\ &\qquad\qquad \, + \, \cdots \, + \, \sin(k+n)a\,\cos na\, ;\\[2\jot] &\mbox{\bf \quad {(ii) }} \tan x \,+\, \frac12\tan\frac{x}2 \, +\,\frac1{2^2}\tan\frac{x}{2^2}\,+\,\cdots\\ &\qquad\qquad \,+\, \frac1{2^n}\tan\frac{x}{2^n}\,.\endaligned}](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-424fd4794b737108f9ceb3b16c37ee85_l3.svg "Rendered by QuickLaTeX.com")
Express
as the cotangent of an angle.
3. Find in terms of the sides of a triangle the area of (i) the triangle, (ii) the inscribed and escribed circles.
The radii of two escribed circles of a triangle are 
and 
, and the distance between their centres is 
. Shew that the area is
Q1 is missing a few command terms…
…and some proper punctuation.
But otherwise is a really nice question.
For anyone wondering about Q3 I learned today that escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two.
Thanks, RF. I had to look it up as well.
I’ve been mostly focusing on Q1 from this paper. Lots of possible approaches, not getting very far (yet) with any of them though. Will post my best progress attempt soon and see if others can carry it from there or give a push in the right direction.
More commonly known as an ‘excircle’ in modern terms.
On question 3: having established the relations K = rs and the formula for the A-exradius r_A = K/(s-a), where K is the area, r is the inradius, and s is the semiperimeter, the second part is computationally tedious but gets you there in the end. I’d be curious to know if there’s a more elegant method.
Thanks, Javert. I’ll look once I level the mountain a little. (But I suck at circles.)