An Indepth Look at The
Unit Circle (Part 1)
We already know that the
Unit Circle has a radius of one. Because of this, we can easily measure the
values of sine, cosine, and tangent. For example, lets plot a point on the Unit
circle, and call it point (x,y). Let’s also call the angle formed by point
(x,y) and the radius angle theta. We already know that sine equals opposite
over hypotenuse, and by definition , the radius of the Unit Circle is one.
Therefore, the sine of angle theta equals y(height of point (x,y)) over 1 or
simply y.
Likewise, we know that
cosine equals the adjacent side over the hypotenuse. Once again, we know that
the radius of the unit circle is one. Therefore, the cosine of angle theta
equals x(length of point (x,y) over one, or simply x.
Based on this
information, we can tell an angle in the unit circle’s sine and cosine values.
Instead of point (x,y), we can look at the coordinates as (cosine theta, sine
theta).
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This becomes even more
useful when we use values (in degrees). Let’s review our 30-60-90 triangle
rule:
Based on what we
learned earlier, we can easily find the cosine and sine of a thirty degree
angle. Imagine the 30-60-90 triangle inside a unit circle.
2 . And of course, the hypotenuse of this triangle is one (it is a radius of the Unit Circle as well.)
Notice that the two
legs of the triangle meet at the point. Recall what we learned about the
coordinates of a point of the Unit circle (reminder : (x,y) --> (cosine, sine)) . Therefore, we now know that
an angle of 30 degrees has a cosine of the sqrt of 3 and a sine of ½.
2
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