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before diving into the world of audio in
the digital domain we need to understand
the fundamentals of audio on the
simplest way to talk about a subject is
to figure out the elemental parts that
make up the subject for a lot of areas
of study this is fairly hard to do since
the makeup is ambiguous and fairly hard
to single out but in the world of audio
it's unmistakably simple simple harmonic
motion the physics behind sound and the
generation of simple harmonic motion is
described in another module on acoustics
but let's talk instead about the
mathematics behind the simple harmonic
motion and more specifically the sine
wave we've all seen the sine wave before
the squiggly wavy shape that you see in
music equalizers a sine wave is the
purest form for you which represents an
exact frequency or a tonal value in fact
sine waves don't exist in isolation in
nature no natural source of sound
generates a singular sine wave but
rather a multitude of sine waves bundled
and added together more surprisingly
it's possible to recreate any sound
including the sound of my own voice
which is adding different sign waves
together although this can get expensive
and absurdly complicated and there are
different synthesis methods to recreate
more complex sounds which you will be
learning about later on sine waves can
truly be thought of as building blocks
of audio like pieces of Lego since sine
waves are quite essential to the study
of audio and since digital systems can
produce single sine waves we need to get
a good grasp of where the sine wave
originates from and we go back to the
humble beginnings of the triangle and
more specifically the right angle
triangle we choose the right angle
triangle here since it has a few
established properties that we can
leverage so what do we know about the
right angle triangle apart from the fact
that one of its sides is at a 90 degree
angle with
the other we know that since the sum of
all the angles of a triangle is a
hundred and eighty degrees
the sum of the two other angles of the
triangle has to be equal to 90 degrees
here angle A plus angle B equal to 90 we
also know that the Pythagorean theorem
is established for right angle triangles
where it states that the sum of the
squares of the sides of a triangle is
equal to the square of the hypotenuse
written out
it's simply X square plus y square equal
to a hypotenuse square what else do we
know we know that there are certain
well-defined but arbitrary ratios that
are conjured up for the right angle
triangle these ratios provide
relationships between the sides and
angles of a right angle triangle and
they're called trigonometric ratios or
trignometric functions there are six and
all but three of them more more commonly
used that's the sine of an angle a which
is the ratio of the opposite side to the
hypotenuse the cosine function which is
the ratio of the adjacent side to the
hypotenuse and then we have the tangent
function which is the ratio of the
opposite side to the adjacent side you
might notice that the sine function is
already described here the end goal of
the module is to describe the sine
function but it's not yet in a form
that's recognizable or consumable for
audio purposes so let's push on you
might wonder at this point of time what
the purpose of these trigonometric
ratios are they are just made up
fractions after all that describes some
relationships however there are several
practical applications these ratios
let's take a simple example let's say
you want to calculate the height of a
building or a tree one way you could
possibly do this is to take a tape
measure climb the tree and measure how
tall it is a less daft of a person would
try to use the power of observational
mathematics to do the same what do we
know about the environment we can
position ourselves at a point P in the
world
and we can look up at the tree and
measure the angle at which we need to
tilt in order to observe the tree from
our position on the ground let's call
this angle theta we can also walk over
to the tree and measure the distance of
the tree from our initial position what
we have here is a rudimentary right
angle triangle where we've determined
one of the angles and the length of the
adjacent side the information we're
lacking is the length of the hypotenuse
which is quite hard to determine in our
case using conventionally means on the
length of the opposite side or the
height of the tree the answer to which
if you're looking for all we need to do
is leverage one of the trigonometric
functions and substitute the values the
tangent function comes to mind tan of
the angle theta is equal to the opposite
side over the adjacent side let's put a
couple of concrete values here say the
distance to the tree is 60 meters and
the angle we are observing it at is 60
degrees if we rearrange the equation we
can easily find the height of the tree
if only we knew the tan of the angle of
60 fortunately I seem to possess the
values of the tan function for a few key
angles and they're tabulated here that's
handy tan of 60 degrees is a root of 3
and root of 3 multiplied by 60 s
approximately a hundred and 4 meters
that's a tall tree it was quite
convenient that we had the tan
conversion table but how did we arrive
at those values tan of theta is said to
be a continuous function over the angle
theta a function in mathematics or in
programming for that matter is a
relationship where you can plug in or
substitute any value for the parameter
that it operates over and you get an
output in each case let's try to find
the values of the sine function for this
we need to simplify the triangle further
any unknowns we can eliminate the better
to do this we turn the circles unit
circles
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