Isn't it just a right angle triangle – the right angle being the base of the V block, the hypotenuse being the centre line X extended to the edge of the V block gives you the Right angle

called a scalene right angled triangle with no equal sides

so the inclusive angle will be 35 degree at the point and 55 degrees at the vertical end

regards Trevor

Edited By Trevorh on 27/03/2019 11:16:52

For those that don't have cad and want to see if they can come up with a formula then these are some random CAD values.

V = 80deg

C = 40deg included angle

X = 32.15deg

I’m sure I’ve seen this solved – look up analysis of conic sections – was in all advanced geometry books many years ago (eg late Victorian text books) – had a quick look at my bookshelves but can’t find the book I was thinking of

yes that works

Sin X = Sin M x Cosec N

Where M is half the value of C and N is half the Value of V from the original question.

So taking those CAD figures I posted earlier

Sin X = Sin 20 x Cosec 40

Sin X = 0.342 x 1.558

Sin X = 0.533

X = 32.21 which allowing for me rounding to 3 decimal places is right.

I did have to go and look up what Cosec was as I only went as far as O level maths

Edited By JasonB on 27/03/2019 14:42:53

Or sin(x) = sin(M/2)/sin(V/2) since csc(x) = 1/sin(x)

Now to prove it!

I'm never quite sure why sec and cosec were ever invented since they are just reciprocals of cos and sin respectively.

Me neither, except reading an old book about navigation suggests an answer: it does sums using haversines, covercosines and other weirdness. The reason seems to be simplification and error reduction of calculations at a time when they were done manually in mid-Atlantic by sea-sick navigators in rough weather. It's quicker and safer to look up cosec x in a table than it is to crunch 1 / sin x, or to look up sin x and then it's reciprocal in two tables. The formula involved are simplified too, for example the Haversine formula is hot at calculating great circle distances. Likely the functions also helped on land, perhaps laying out curves on railway lines.

Other examples of 'labour saving' trig functions include Versine, Coversine, Vercosine, Secant, Cosecant, Exsecant, Excosecant, Cotangent and probably others. Once you have a computer, calculator or even a slide rule, the need for them largely disappears, I guess. I've never come across them in real life!

Dave

In the days of solving math using pencil & paper it was simpler to multiply by the inverse function than to divide by the function. Some of the early mechanical calculators were not capable of division hence the need for the inverse functions.

The thing that threw me was that the round disk of the wide part of the cone, when angled forward starts making contact with the V block nearer and nearer to the east and west positions. Looking from the front, the disk starts to become elliptical the more it leans.

Brian, just click on the link I posted and you can scroll upto the front cover and read the title etc.

For the benefit of others, the book is titled " Mathematics at Work" . This reference was taken from the 4th Edition

The author is Holbrook L Horton and the book is published by the Industrial Press in New York

Brian