Transcript The Unit Circle

4
Trigonometric Functions
Copyright © Cengage Learning. All rights reserved.
4.2
Trigonometric Functions:
The Unit Circle
Copyright © Cengage Learning. All rights reserved.
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
4
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
Unit circle
as shown in Figure 4.18. It is called
the unit circle because it has a radius
Figure 4.18
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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 a
circumference 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
cosecant secant
tangent
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
cont’d
t =  /6 corresponds to the point
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Example1(b) – Solution
cont’d
t = 5 /4 corresponds to the point
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Example1(c) – Solution
cont’d
t =  corresponds to the point (x, y) = (–1, 0).
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Example1(d) – Solution
cont’d
Moving clockwise around the unit circle, it follows that
t = – /3 corresponds to the point
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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
b. Because
you have
you have
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Example 2 – Using the Period to Evaluate Sine and Cosine
c. For
cont’d
because the function is odd.
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