MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

Trigonometry Today

Now that we know how Ptolemy introduced trigonometry to the world, let's turn the focus to something more familiar - sine, cosine, and the unit circle.  For us, our circle has a radius of 1 unit.  However, unlike Ptolemy, our trigonometry today goes a couple of steps past anything Ptolemy ever created.

First, our unit circle is placed on the Cartesian plane, as seen in the diagram at left with the center of the circle at the origin (0, 0).  Creating the unit circle in this manner allows us to identify points along the circle's edge. To identify a point on the circle, we draw a radius from the point to the circle's center, making an angle with respect to the x-axis, which he designate as the Greek letter theta - θ.  The coordinates of our point are (x, y) making is easier for us to label points on the plane.  However, as most know, there is a method to labeling points in trigonometry.

When we label points on the circle, we create right triangles in order to represent points using two famous trigonometric functions - sine and cosine.  In a right triangle, sin θ is the ratio of the length of the side opposite θ divided by the hypotenuse of the triangle.  For cos θ, we divide the length of the side of the triangle adjacent to θ by the hypotenuse. In the figure at the right, which is an exact replica of the above figure, we find that:

Recalling from our discussion of Ptolemy, we learned that Ptolemy computed two of his Crd functions rather easily - Crd 90º and Crd 120º.  In our trigonometry, we can easily find the sine and cosine of four angles almost immediately.  These angles happen to be at the points of the intersections of the x and y axes.  For example, if we wanted to know the sin 90º, we would find the point that makes a right angle with the x-axis.  All angles are made with respect to the x-axis.  According to the diagram, it is the point (0, 1).  This means that since y =1, and y = sin θ, then sin 90º = 1.  Likewise, cos 90º = 0.  We can perform this same procedure for these other angles - 180º, 270º, and 360º - the other intersection points along the circle.

We also have a pair of special triangles that allow us to compute even more sine and cosine functions.  Most of the time, they are  referred to as the 30-60-90 right triangle and the 45-45-90 right triangle.  With these two triangles, we are now able to calculate the sine and cosine of 30º, 60º, and 45º in the first quadrant (Q I at right) of the Cartesian plane.  Using the two triangles in the figure at the right (and imagining them placed in the unit circle) we can see that the sin 30º = ½ (opposite side is 1, hypotenuse is 2), sin 60º = √3/2 (adjacent side is √3 units, hypotenuse is 2), and sin 45º = 1/√2.  Similarly, cos 30º = √3/2, cos 60º = ½, and the cos 45º = 1/√2. However, this is only in the first quadrant!  Thus, we can also find other angles in other quadrants such as 120º, 150º, 135º, 210º, 240º, 225º, 300º, 330º, and 315º using the same triangles at the right.

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