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David Terr
Ph.D. Math, UC Berkeley

>> 4.3. Right Triangle Trigonometry

4.2.Trigonometric Functions: The Unit Circle

In the previous section, we defined the measure of an angle with the unit circle, a circle of radius 1 centered at the common endpoint of the two rays of the angle. In this section, we define the six trigonometric functions of an angle. For this purpose, we continue to use the unit circle. We will also find it convenient to set up a coordinate system for labeling points as shown below.

Refering to the above diagram, we now define the trigonometric functions sine and cosine as follows: The sine of the angle θ, denoted as sin θ, is the y-coordinate of the point P shown above, and the cosine of the angle θ, denoted as cos θ, is the x-coordinate of P. This is shown in Figure 4.2.1 below.

Figure 4.2.1: Definitions of Sine and Cosine

It is clear from Figure 4.2.1 that for all angles θ between 0 and π/2, both sin θ and cos θ are between 0 and 1. It is also clear that sin 0 = 0, cos 0 = 1, sin π/2 = 1, and cos π/2 = 0. In fact, it is easy to see from the figure that we have the following values of cos θ and sin θ for angles θ which are multiples of π/2:

θ sin θ cos θ
0
0
1
π/2
1
0
π
0
-1
3π/2
-1
0

As mentioned eariler, there are four other trigonometric functions, namely the tangent, cotangent, secant, and cosecant. Like the sine and cosine, these functions also have simple geometric interpretations in terms of the unit circle. We illustrate these with the following diagram, along with sine and cosine.

Figure 4.2.2: Definitions of All Six Trigonometric Functions

As can be seen from the figure, the tangent of the angle θ, abbreviated as tan θ, is the y-coordinate of the point R of the line QR, also known as the tangent line to the unit circle at the point Q = (1,0). (Tangent means touching.) Similarly, the cotangent of the angle θ, abbreviated as cot θ, is the x-coordinate of the point T of the line ST, the tangent line to the unit circle at the point S=(0,1).

The only trigonometric functions not explicitly shown on the diagram are the secant and cosecant, though these have simple geometric interpretations as well. The secant of θ, abbreviated as sec θ, is the length of the line segment OR, while the cosecant of θ, abbreviated as csc θ, is the length of the line segment OT. One should be careful with these definitions however, because for some angles, the secant and cosecant can become negative!

It should be noted that the four functions tan θ, cot θ, sec θ, and csc θ can all be simply expressed in terms of the two more basic trigonometric functions sin θ and cos θ. Specifically we have the following identities, which we will prove later.

• (4.2.3) tan θ = sin θ / cos θ
• (4.2.4) cot θ = 1 / tan θ = cos θ / sin θ
• (4.2.5) sec θ = 1 / cos θ
• (4.2.6) csc θ = 1 / sin θ

>> 4.3. Right Triangle Trigonometry