4.4 Trigonometric Functions of Any Angle
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4.4
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Trigonometry
Trigonometric Functions of Any Angle
What you should learn
? Evaluate trigonometric
functions of any angle.
? Use reference angles
to evaluate trigonometric
functions.
? Evaluate trigonometric
functions of real numbers.
Why you should learn it
You can use trigonometric
functions to model and solve
real-life problems. For instance,
in Exercise 87 on page 319, you
can use trigonometric functions
to model the monthly normal
temperatures in New York City
and Fairbanks, Alaska.
Introduction
In Section 4.3, the definitions of trigonometric functions were restricted to acute
angles. In this section, the definitions are extended to cover any angle. If is an
acute angle, these definitions coincide with those given in the preceding section.
Definitions of Trigonometric Functions of Any Angle
Let be an angle in standard position with x, y a point on the terminal side
of and r x2 y2 0.
sin
y
r
y
tan ,
x
cos
x
r
x0
x
cot ,
y
x0
r
csc ,
y
r
sec ,
x
y
(x , y)
y0
r
y0
¦È
x
Because r x 2 y 2 cannot be zero, it follows that the sine and cosine
functions are defined for any real value of . However, if x 0, the tangent and
secant of are undefined. For example, the tangent of 90 is undefined.
Similarly, if y 0, the cotangent and cosecant of are undefined.
Example 1
Evaluating Trigonometric Functions
Let 3, 4 be a point on the terminal side of . Find the sine, cosine, and
tangent of .
Solution
Referring to Figure 4.36, you can see that x 3, y 4, and
r x 2 y 2 3 2 42 25 5.
James Urbach/SuperStock
y
(?3, 4)
So, you have the following.
sin
y 4
r
5
cos
x
3
r
5
tan
y
4
x
3
4
3
r
2
1
?3
FIGURE
?2
4.36
?1
¦È
x
1
Now try Exercise 1.
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11:05 AM
Page 313
Section 4.4
y
¦Ð 0
x
x ................
................
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