The Unit Circle
The Unit Circle is
a tool used in
understanding
sines and cosines
of angles found in
right triangles. It
is so named
because its radius
is exactly one unit
in length, usually
just called "one".
The circle's center
is at the origin,
and its
circumference
comprises the set
of all points that
are exactly one
unit from the
origin while lying
in the plane.
To use the unit circle, we put the vertex of an
angle at the center of the circle with the first
side of the angle extending along the x-axis
from the vertex toward the right. The other
side of the angle will fall somewhere
(depending on the size of the angle) around
the circle, counter-clockwise from the first
side. Positive angles are always measured
counter-clockwise from the zero degree point
on the unit circle.
Next, we draw a
vertical line from
the point where the
second side of the
angle meets the
circle to the x-axis
(the vertical line
must form a right
angle with the xaxis).
The circle's radius being one unit in length is important now, because
the sine and cosine are defined as the opposite or adjacent
(respectively) over the hypotenuse. The hypotenuse equaling one
simplifies the calculations greatly! On the unit circle, we need only
measure the opposite or adjacent side of the triangle for an accurate
measurement of the sine or cosine.
In PreCalculus class,
the angles on
the unit circle
really have to
be memorized.
But we might
look at the
"square root of
three divided
by two" and
wonder what
this really
means. If the
values were a
little more
meaningful,
they might be
more
memorable!
Cosine and
sine values in
this diagram
have been
converted into
some perhaps
more familiar
forms. A little
algebra reveals
that the 30o
angle has a
cosine of
and a sine of
.
Perhaps this
way of
looking at the
sine and
cosine values
will be helpful
in your
memorization.
But as long as we're looking at it this
way, please consider the Pythagorean
theorem. The cosine of 30o is
(the
width of the blue box) and the sine is
(the height of the yellow box) and
these are two sides of a right triangle!
The Pythagorean theorem says that if
we add the squares of these two sides
together, we get the square of the
hypotenuse.
gives us 3/4 (the area of the
blue box), and
is 1/4 (the area
of the yellow box). Well, of course
3/4 plus 1/4 is 1, but here's the
important part: sine squared plus
cosine squared of a given angle
always equals 1.
Here's how
sine and
sine
squared,
and cosine
and cosine
squared
look for
various
angles.
Notice the
hypotenuse
(the
radius) is
always 1,
so the
hypotenuse
squared
never
changes
size.
How do
the sine
and cosine
functions'
graphs
relate to
the angles
on a unit
circle?
Click here!
You need to
memorize the
unit circle, but
if those other
ways don't
seem to do it
for you, try
this:
Remember that
each of these
fractions is
"over two", and
each numerator
is under a
radical;
Starting at the
0o point and
looking at sine,
the numbers
under the
radicals are: 0,
1, 2, 3, 4 as
you work your
way around
counterclockwise.
Once you get
to the 90o
point, go back
down to 0o
using the same
pattern, only
now for cosine.
Watch out for
the negative
signs in the
other
quadrants, but
the numbers
are all the
same.
Remember:
Denominator is
2, Numerator is
under a radical
(yes, you can
skip the radical
with 0 and 1 if
you wish).
Okay,
that's fine
for sine
and cosine,
but what
about
tangent?
Tangent is
the
quotient of
the sine
over the
cosine, and
sine and
cosine are
between
positive
and
negative 1.
Division
problems
like this
are
sometimes
difficult.
Well, back
to our Unit
Circle!
With your
angle in
standard
position,
it's easy to
see the
tangent if
you extend
the
hypotenuse
to the right
until it
touches the
x = +1
line. The
tangent is
the same
as the y
value
where the
hypotenuse
intersects x
= +1.
Note: even
if your
angle has a
negative
cosine, you
still have
to extend
to the
right.