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4.1. Angles and Their Measure

This chapter is devoted to trigonometry, which is the study of the geometry of angles and functions devoted to them, such as sine, cosine, and tangent. In this section we define an angle as well as its measure, both in degrees and in radians.

An * angle* is two rays with a common endpoint. A diagram of an angle is shown below.

So what do we mean by the measure of an angle? To measure an angle, we draw a circle of radius 1, better known as a * unit circle*, centered around the common endpoint O of the rays of the angle, as shown in Figure 4.1.1 below. The

*of the angle in*

**measure***is then defined as the length of the arc of the circle between the points P and Q of intersection of the circle and the rays. The greek letters θ (theta) and φ (phi) are commonly used to denote angular measure. As one can see in the figure, the line segments OP and OQ each have length 1, while the arc PQ has length θ, which we then call the measure of the angle ∠POQ.*

**radians**Figure 4.1.1: Angular Measure Diagram

As an example, consider a right angle, as shown in Figure 4.1.2. What is its measure in radians? Since the corresponding arc is one-quarter of the unit circle, whose total arc length is 2π, we see that the measure of a right angle is 2π/4 = π/2 radians.

It should be noted that the unit of radians is not always used explicitly but implied. Thus, for instance, one often says the measure of a right angle is π/2.

Figure 4.1.2: Right Angular Measure

Besides radians, angles are commonly measured in another unit called * degrees*. The measure of an angle in degrees is equal to 180/π times its measure in radians. Thus, the measure of a right angle is (180/π)(π/2) = 90 degrees, also written as 90°, and the measure of a full circle is (180/π)(2π) = 360 degrees, or 360°. To convert from degrees to radians, we multiply by π/180. Thus, for instance, an angle of 60° is equal to (60)(π/180) = π/3 radians.

**Example 1: **Convert the following angular measures from degrees to radians.

- (a) 45°
- (b) 120°
- (c) 180°
- (d) 1°

**Solution: **

- (a) We have 45° = (45)(π/180) = π/4 radians.
- (b) We have 120° = (120)(π/180) = 2π/3 radians.
- (c) We have 180° = (180)(π/180) = π radians.
- (d) We have 1° = (1)(π/180) = π/180 radians.

**Example 2: **Convert the following angular measures from radians to degrees. (Note that the use of radians is implied here, though not explicitly used.)

- (a) π
- (b) π/6
- (c) 3π/4
- (d) 1

**Solution: **

- (a) We have π radians = (π)(180/π) = 180°.
- (b) We have π/6 radians = (π/6)(180/π) = 30°.
- (c) We have 3π/4 radians = (3π/4)(180/π) = 135°.
- (d) We have 1 radian = (1)(180/π) = 180/π degrees ≈ 57.3°.