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Chapter 1 | Functions and Graphs
1.3 | Trigonometric Functions
Learning Objectives 1.3.1 Convert angle measures between degrees and radians. 1.3.2 Recognize the triangular and circular definitions of the basic trigonometric functions. 1.3.3 Write the basic trigonometric identities. 1.3.4 Identify the graphs and periods of the trigonometric functions. 1.3.5 Describe the shift of a sine or cosine graph from the equation of the function.
Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. Radian Measure To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle θ , let s be the length of the corresponding arc on the unit circle ( Figure 1.30 ). We say the angle corresponding to the arc of length 1 has radian measure 1.
Figure 1.30 The radian measure of an angle θ is the arc length s of the associated arc on the unit circle.
Since an angle of 360° corresponds to the circumference of a circle, or an arc of length 2 π , we conclude that an angle with a degree measure of 360° has a radian measure of 2 π . Similarly, we see that 180° is equivalent to π radians. Table 1.8 shows the relationship between common degree and radian values.
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