Re: On the uniformity of states
Compare this passage from Lockhart's Lament:
"Of course it is far easier to test someone's knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning. Even if we agree that a basic common vocabulary for mathematics is valuable, this isn't it. How sad that fifth-graders are taught to say "quadrilateral" instead of "four-sided shape," but are never given a reason to use words like "conjecture," and "counterexample." High school students must learn to use the secant function, 'sec x,' as an abbreviation for the reciprocal of the cosine function, '1 / cos x,' (a definition with as much intellectual weight as the decision to use '&' in place of "and." ) That this particular shorthand, a holdover from fifteenth century nautical tables, is still with us (whereas others, such as the "versine" have died out) is mere historical accident, and is of utterly no value in an era when rapid and precise shipboard computation is no longer an issue. Thus we clutter our math classes with pointless nomenclature for its own sake."
with this site for Indian students (http://www.ilovemaths.com/3allied.asp):
"Without using trigonometric tables, evaluate (34-43):
(cos0° +sin45° +sin30°).(sin90° -cos45° +cos60°) sin² 45° -tan² 60° + cos² 90° sin 23°/cos 67° cosec 31°/ sec 59° sin² 38° -sin²52° sin 18° -cos 72° sin 36° sec 54° +cos 24° cosec 66°"
and this Yahoo Answers question from some Argentinian student (auto-translated from Spanish):
"NEED HELP IN TRIGONOMETRIA!? I need to know how to solve this: Given the triangle ABC rectangle in A, with the legs 6 and 8 cm find the six trigonometric functions of the angle B. Base: 8 cm H: 6 cm Graciaaas
Best Answer - Chosen by Voters
Taking the angles of the chart as follows: http://sketchtoy.com/53599248 Then the angle 'b' is: sin b = 8/10 = 0.8 cos b = 6/10 = 0.6 so b = 8/6 cosec b = 10/8 sec b = 10/6 cotan b = 6/8"
and this from your very own Tanzania (forgive the mangled copy-pasting):
"Prove the given trigonometric relation 22seccsctancotseccscθθθθθθ+−=
- Prove the given idenity
221cotcscθθ+=
- Solve forx, (a) (b) 23tan10x−=1sincos4cossin1xxxx++=+ (c) 3tan()302x+="