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294 Chapter 4 Trigonometry
Example 1
Evaluating Trigonometric Functions
You can review dividing
fractions and rationalizing
denominators in Appendix A.1
and Appendix A.2, respectively.
Evaluate the six trigonometric functions at each real number.
a. t b. t 5 c. t 0 d. t
6
4
Solution
For each t-value, begin by finding the corresponding point x, y on the unit circle. Then
use the definitions of trigonometric functions listed on page 293.
a. t corresponds to the point
6
cos
sin 6 y 1 2
6 x 3
2
tan 6 y x 12
32
b. t 5 corresponds to the point
4
5
sin
4 y 2 2
5
cos
4 x 2 2
5
tan
4 y x 22
x, y
3
2 , 1 2 .
1
3 3
3
x, y 2
2 , 2 2 .
22 1 sec 5
c. t 0 corresponds to the point x, y 1, 0.
csc
6 1 y 1
12 2
sec
cot
6 1 x 2 3 23
3
6 x y 32
12
3
csc 5
4 1 y 2
2 2
4 1 x 2
2 2
cot 5
4 x y 22
22 1
sin 0 y 0
csc 0 1 is undefined.
y
cos 0 x 1
sec 0 1 x 1 1 1
tan 0 y cot 0 x is undefined.
x 0 1 0 y
d. t corresponds to the point x, y 1, 0.
sin y 0
csc 1 is undefined.
y
cos x 1
sec 1 x 1
1 1
tan y cot x is undefined.
x 0
1 0 y
Now try Exercise 23.