What is Trigonometry? B R Sitaram Zeal Education

What is Trigonometry?  Few branches of mathematics confuse and scare students more than trigonometry  The reasons are many: The number of confusing names to be remembered: sin, cos, tan, cot, cosec, sec! Is sin Opposite/Hypotenuse or is it Adjacent/Hypotenuse?

What is Trigonometry? The large number of identities: between functions, addition & subtraction of angles, multiple angles, … (Largely) meaningless exercises: e.g. show that: cos 35/sin 55 + tan 27 tan 63/sin 30 – 3 tan 2 60 = -6 SO WHAT??? The way the subject is introduced, with no connection to other branches of maths.

What is Trigonometry?  This presentation is aimed at: Showing the connection of geometry and trigonometry Showing why right angled triangles are chosen for introducing sin, cos, etc Showing the importance of the addition formulae for creating tables of trigonometric functions

What is Trigonometry?  All of trigonometry is based on one concept and one theorem  Concept: Similarity!  Two figures are similar if one is a scaled down version of another!  Concept of similarity crucial for all modelling: to make an accurate model of the Parthenon, the model must be similar to the original!

What is Trigonometry?  Considered to be so basic an idea (along with congruence), it is assumed to be “obvious” by Euclid!  The Theorem: If in two triangles, the angles of one equal the angles of another, the triangles are similar

What is Trigonometry?  What does this mean?  Consider the two triangles shown here and assume that A = P, B = Q and C = R. Then, a/p = b/q = c/r!

What is Trigonometry?  Consequence: a/b = p/q, a/c = p/r and b/c = q/r!!  If the three angles of the triangle are prescribed, the ratios of the three sides are fixed!

What is Trigonometry?  Hence, can build a table: you tell me the three angles of the triangle, I will tell you the ratios of the three sides!  Q: Do you need three angles? Can we manage with fewer?  A: Certainly, two are adequate, as third angle is known as soon as we know two!  Can we reduce it further? Say to one angle only?

What is Trigonometry?  Sure, here’s how: Drop a perpendicular from A. Since I know B, and the right angle at D, I know the ratios for triangle ABD: a 1 /d, a 1 /c, c/d. Similarly for ACD: I know a 2 /b, a 2 /d, b/d Hence, we know ratios for ABC: a/b, a/c, b/c!

What is Trigonometry?  We can therefore construct a new table: Give me one angle of a right angled triangle, I will give you the ratios of the three sides.  Use this info to get the ratios of the sides for ANY triangle!  The ratios for a right angled triangle: sin, cos, tan, sec, cosec and cot!  Depend on only one angle!!!

What is Trigonometry? How do we make the table?  In principle, very simple. Draw the triangle to ANY SCALE, measure the sides!  For example, if angle = 40: Take a convenient length for base, draw triangle with angles 40, 50 and 90.  Measure the three sides and find ratios.

What is Trigonometry?  See example on the right: all ratios known!  Any other triangle with same angles will have same ratios!  Note: Triangle drawn using Geogebra, copied to Paint, measured in Pixels and hypotenuse calculated using Pythagoras

What is Trigonometry?  Tedious to do this for each angle.  Use addition formula! Relates ratios for  A and  B to  A+B!  Hence relate ratios for A to 2A and hence to A/2.  We know ratios for 60 (half an equilateral triangle) and 45 (isosceles) from Euclid’s Geometry.

What is Trigonometry?  From 60, we know ratios for 30, 15, 7.5, 22.5 (15 + 7.5), etc.  Hence complete tables can be built for multiples of a particular unit.  First such tables calculated by Hipparchus ( BCE) and Ptolmey ( CE). Aryabhatta (476–550 AD) calculated ratios in increments of 3.75 (half of 7.5)

What is Trigonometry?  This method is no longer used to build tables, better methods used.  BUT, in principle, all you need to know are the ratios of some special triangles and the addition formulae!

What is Trigonometry? TRIGONOMETRY  SIMILARITY OF TRIANGLES! TRIGONOMETRY TABLES  ADDITION FORMULAE AND RATIOS FOR SPECIAL TRIANGLES.

What is Trigonometry?  Notes:  For the addition formula, see my video “Trigonometry formulae for addition of angles” on YouTube.  The similarity theorem is NOT valid for other polygons. For example, all rectangles have all 4 angles equal, but the ratios of the sides is not fixed! You need more conditions for similarity!