



If you punch sin 45° into your calculator,




it probably comes back with
0.70710678 


( Maybe a few more or less decimal places) 


If you punch sin 135° into your calculator 


you get the exact same number 0.70710678! 


Why? 





To answer that question, we have to draw a right triangle 


and drop an
X Y coordinate grid ( better known as a graph) on it. 


Ouch! I hope the triangle wasn't hurt. 











Now the thing that happens, is that the side of the triangle at the bottom 


goes from
the origin (0,0 point) on the left, to some X value on the right. 


We'll call that X value:
X (How original). 


That means that the triangle side on the bottom 


goes from the Joint (0, 0) to the point (X,0). 


Also the side on the right goes from that point (X,0) 


to
a point with the same X value, and some Y value, 


we'll call the point (X, Y). 











The length of the side on the bottom is X, 


and the length of the side on the right is Y. 








OK, The last side of the triangle is the hypotenuse. 


For reasons
that will make sense in a little while, 


we are going to call the
length of the hypotenuse 1. 


So we have: 








Now let's look at the pointy little angle on the left. 


What's the sine of this angle (call it "A"). 











What is the Cosine of A? 











So because the
hypotenuse is 1, 


the length of the side on the right of the triangle 


is the value of the cosine of the pointy little angle on the
left. 





And because the hypotenuse is 1, 


the length of the side on the bottom
of the triangle 


is the value of the sine of the pointy little angle on the left. 





This is a
very special way to arrange a triangle. 


Since the
hypotenuse is equal to 1 


the length of the triangle side running
in the X direction 


is always equal to the sine of the angle at
the origin of the graph. 





In this
arrangement, 


the cosine of the angle at the origin is always 


equal to the length of the triangle running in the Y direction. 





This
arrangement is so special that math types call it "STANDARD
POSITION" 





It follows
directly from this 


that the tangent of the angle at the origin 


is equal to the sine (Y) divided by the cosine (X) 





You can
think of the hypotenuse like the hour hand on a watch. 


At two
o'clock, you have this: 





Since
there are 90°
from straight up to straight out to the right, 


each hour on a 12
hour clock face is equal to 30°. 


We are looking at the angle from the X axis (3
O'clock) 


to one
hour up at 2 o'clock where the hypotenuse is. 


That makes the
angle 30° 





We can say that angle A =
30°. 


We can
look up the sine of 30°. 


It is 0.5 


We know
that the sine of an angle is equal to 


the side opposite divided by the
hypotenuse 


and the hypotenuse in this case is 1. 


That means: 











0.5 = Y 





This is just what we expected. 





We know that the Cosine of an angle 


is equal to the side adjacent
divided by the hypotenuse 


and in this case the hypotenuse is 1. 


That means
... 











0.866 = X 





What we have done is say
... 


If we think about a triangle where the
hypotenuse is 1, 


we really don't need the triangle at all. 


We
can have sines, cosines, and all of the rest. 


We can go back and
check it on a triangle if we want, 


but we really don't have to. 





Those might sound like a trip into "WHO CARES LAND," 


but most uses of uses of trig do not
deal with triangles. 


This is the path that lets us step away from the triangles where trig was born 


to
the places where we have many other uses for it. 





Let's do a couple more of these just to make sure it makes sense. 





Example: 





Let's try 1
O'clock: 





Two hours is equal to 60° so: 


The sine 60° equals 0.866 so the Y length on our triangle is 0.866 


The cosine of 60° equals 0.5 so the X length on our triangle is 0.5 


The tangent of
60° equals our Y length divided by our X
length 


so the tangent of 60° equals 1.732 





Example: 





Now a tricky one. Let's try 12 o'clock: 





12 O'clock means three hours from the X axis, 


and at 30°
per hour that means 90°. 











OK,
We might have a problem. 


At 12 o'clock we don't have a triangle. 


The length of the X side is zero and the Y side is the same as
the hypotenuse, 





That's
true, but the answer is: 





Who
cares! 





We were
just using the triangles as a helper. Just look at the X and Y
values. 





What's
the value of X? ZERO 





So
if the Cosine of an angle is equal to X, What's the cosine of 90°?
ZERO! 





So
what's the sine of 90°? 





Since
at 12 O'clock the hour hand runs along the Y axis, 


the Y length
is 1. That means: 





The sine
of 90° is 1 





That
one was a little tricky. 


The next one is the real doorway to the
world of no triangles. 





Example: 





Let's
try 11 O'clock: 








Which
if we keep on doing things the way we have, 


is an angle of 120°
(4 hours back from 3 O'clock at 30°
per hour) 











The
problem starts when we make the triangle from this. 


To get back
from the hypotenuse to the X axis 


and then back from there to
the origin (0,0) point, 


the triangle we get is: 











Which
has nothing to do with a 120°
angle. 


OR
DOES IT? 





This
is the moment we've all been waiting for! 


We'll
OK, this is the moment that I have been waiting for 


and you have
been humoring me about. 





Let's
compare the 60°
angle (1 O'clock) and the 120°
angle (11 O'clock) 











And
punch up some values on a calculator: 





Angle:
60° 

Angle:
120° 
Sine
= Y = 0.866 

Sine
= Y = 0.866 
Cosine
= X = 0.5 

Cosine
= X = 0.5 
Tangent
= ^{Y}/X = 1.732 

Tangent
= ^{Y}/X = 1.732 






Except
for some minus signs here and there, they're the same! 





Now
let's compare 240°
and 300°: 











And
the values for these are: 





Angle:
240° 

Angle:
300° 
Sine
= Y =  0.866 

Sine
= Y =  0.866 
Cosine
= X =  0.5 

Cosine
= X = 0.5 
Tangent
= ^{Y}/X = 1.732 

Tangent
= ^{Y}/X = 1.732 






Again
we get the same numbers 


except sometimes the sign of the number
is different. 





Let's
take a look at what goes on here. 


The
sine is the value of Y. At small angles, Y is small. 











As
the angle gets bigger towards 90°,
Y gets bigger too. 











When
we get to 90°
the triangle disappears, Y is equal to 1, 


so the sine of 90°
is equal to 1. 











As
the angle gets bigger than 90° 


Y, and so also the sine, get smaller than 1. 











As
the angle gets to 180°
Y, and so also the sine, go to 0. 











As
the angle gets to be more than180° 


Y, and so also the sine, become negative. 











The
angle grows to be 270°,
where the triangle becomes a line again. 


Y
is the same length as the hypotenuse again, 


but this time it is
going in the negative Y direction. 


So
the sine of 270°
is 1. 











After
270°, Y is
still negative, but it is getting shorter. 


That is, it's a
smaller negative number. 











When
we get to 360°,
Y has shrunk back to zero, so the sine of 360° is 0. 











360°
is really the same place as 0°,
so the sine of 0°
is also 0. 


Just
because we went all the way around the circle 


does not mean we
have to stop. 


We can just keep going around and around. 











405°,
for example, is the same place as 45°, just once more around
the circle. 


That
means the values we get for all the trig functions 


(Sine,
Cosine, Tangent, etc) 


will be the same for 405°
as they were for 45°. 


In fact, and here's the big idea, 


if you
start at any angle and add any number of 360°'s to it 


the trig
ratios will all be the same. 





One
last point. 


Angles measures in the counter clockwise direction
(like we have been doing) 


are considered positive. 


Angles
measured in the clockwise direction are considered negative. 





That
means that an angle like 120°
is the same angle as 240° 


AND will have all the same values of
it's trig functions. 











copyright 2008 Bruce Kirkpatrick

