The Unit Circle
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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. |
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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.
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| 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. | |||
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In Pre-Calculus 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.
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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! |
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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 counter-clockwise. 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. 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. |
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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.