Larson 4.2

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4.2 Trigonometric Functions: The Unit Circle

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What You Should Learn

• Identify a unit circle and describe its relationship to real numbers.

• Evaluate trigonometric functions using the unit circle.

• Use domain and period to evaluate sine and cosine functions and use a calculator to evaluate trigonometric functions.

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The Unit Circle

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The Unit Circle

The two historical perspectives of trigonometry incorporate different methods of introducing the trigonometric functions.

Our first introduction to these functions is based on the unit circle.

Consider the unit circle given by

x2 + y2 = 1

as shown in Figure 4.18.

Figure 4.18

Unit circle

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The Unit Circle

Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.19.

Figure 4.19

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The Unit Circle

As the real number line is wrapped around the unit circle, each real number t corresponds to a point

(x, y)

on the circle.

For example, the real number 0 corresponds to the point (1, 0). Moreover, because the unit circle has acircumference of 2, the real number 2 also corresponds to the point (1, 0).

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The Unit Circle

In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula

s = r

(with r = 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle given in radians.

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The Trigonometric Functions

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The Trigonometric Functions

The coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t.

sine cosine tangent

cosecant secant cotangent

These six functions are normally abbreviated sin, cos, tan, csc, sec, and cot, respectively.

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The Trigonometric Functions

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The Trigonometric Functions

In the definitions of the trigonometric functions, note that the tangent and secant are not defined when x = 0.

For instance, because t = /2 corresponds to (x, y) = (0, 1), it follows that tan( /2) and sec( /2) are undefined.

Similarly, the cotangent and cosecant are not defined when y = 0.

For instance, because t = 0 corresponds to (x, y) = (1, 0), cot 0 and csc 0 are undefined.

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Example 1 – Evaluating Trigonometric Functions

Evaluate the six trigonometric functions at each real number.

a. b. c. d.

Solution:

For each t-value, begin by finding the corresponding point (x, y) on the unit circle. Then use the definitions of trigonometric functions.

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Example1(a) – Solution

t = /6 corresponds to the point

cont’d

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Example1(b) – Solution

t = 5 /4 corresponds to the point

cont’d

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Example1(c) – Solution

t = corresponds to the point (x, y) = (–1, 0).

cont’d

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Example1(d) – Solution

Moving clockwise around the unit circle, it follows that t = – /3 corresponds to the point

cont’d

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Domain and Period of Sine and Cosine

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Domain and Period of Sine and Cosine

The domain of the sine and cosine functions is the set of all real numbers.

To determine the range of these two functions, consider the unit circle shown in Figure 4.22.

Figure 4.22

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Domain and Period of Sine and Cosine

Because r = 1 it follows that sin t = y and cos t = x.

Moreover, because (x, y) is on the unit circle, you know that –1 y 1 and –1 y 1.

So, the values of sine and cosine also range between –1 and 1.

–1 y 1 –1 x 1

and

–1 sin t 1 –1 cos t 1

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Domain and Period of Sine and Cosine

Adding 2 to each value of in the interval [0, 2] completes a second revolution around the unit circle, as shown in Figure 4.23.

Figure 4.23

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Domain and Period of Sine and Cosine

The values of sin(t + 2) and cos(t + 2) correspond to those of sin t and cos t.

Similar results can be obtained for repeated revolutions (positive or negative) around the unit circle. This leads to the general result

sin(t + 2 n) = sint and cos(t + 2 n) = cost

for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic.

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Domain and Period of Sine and Cosine

It follows from the definition of periodic function that the sine and cosine functions are periodic and have a period of 2. The other four trigonometric functions are also periodic.

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Domain and Period of Sine and Cosine

A function f is even when f (–t) = f (t)

and is odd when f (–t) = –f (t)

Of the six trigonometric functions, two are even and four are odd.

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Example 2 – Using the Period to Evaluate Sine and Cosine

a. Because you have

b. Because you have

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Example 2 – Using the Period to Evaluate Sine and Cosine

c. For because the function is odd.

cont’d