Transcript What is Trigonometry?

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 tan2 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:
a1/d, a1/c, c/d.
 Similarly for ACD: I know
a2/b, a2/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 (180-125 BCE) and
Ptolmey (90-180 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!