 # why we use unit circle to find trigonometric functions???????

10 years ago
cuz,it is most simple and efficient way to do it
10 years ago

Unit Circle Trigonometry

First, we need to understand the unit circle.  As the name  implies, it is a circle where the radius is 1 (one unit).  The unit circle equation is x² + y² = 1 and the graph looks like this: .

Now, if we start at the point (1, 0) and walk a distance t around the circle, we will arrive at a point (x, y) represented by the blue point on the circle in the figure below.  The distance traveled, t, is shown in red.  At the right of the circle is a red line segment that ends in a blue point that is exactly the same length as the red arc ending in the blue point. We are interested in the (x, y) coordinates of the point corresponding to going around the circle a distance of t.

There are six trigonometric functions: sine (abbreviated sin), cosine (abbreviated cos), tangent (abbreviated tan), cosecant (abbreviated csc), secant (abbreviated sec), and cotangent (abbreviated cot).

Here''s how the trigonometric functions are defined.  Let t be the distance traveled around the unit circle ending at the point (x, y):
sin(t) = y,
cos(t) = x,
tan(t) = y/x so long as x is not 0,
csc(t) = 1/y, so long as y is not 0,
sec(t) = 1/x so long as x is not 0,
cot(t) = x/y so long as y is not 0.
Note that csc is the reciprocal of sin, sec is the reciprocal of cos, and cot is the reciprocal of tan.

Here is an interactive website that helps you explore the relationship between the (x, y) point on the circle and the value of the sine, cosine, tangent, etc.: Six Trig Functions.  To use the applet, move the mouse over the red dot (point) on the circle.  Then click and drag the point around the circle to see the measure of the angle change as well as the sin & cos.  You can have the applet display the other trig functions (cot, csc, etc.) as well.

A little mneumonic (memory aid) that I use to keep sine and cosine straight is that x comes before y in the alphabet and cosine comes before sine in the dictionary.  Therefore the cos(t) = x and sin(t) = y.  If it helps you remember, please feel free to use it.

There are some important angles to know by heart (memorize).  They are summarized in the figure below. Here''s how to read the figure.  Look at the point corresponding to t = π/3.  The x-coordinate is 1/2 = 0.5 and the y-coordinate is sqrt(3)/2.  Therefore cos(π/3) = 0.5 and the sin(π/3) = sqrt(3)/2. 