Textbook Chapter 1

Section 1.6 Trigonometric Functions 47
Angle Convention: Use Radians
From now on in this book it is assumed
that all angles are measured in radians
unless degrees or some other unit is
stated explicitly. When we talk about
the angle p3, we mean p3 radians
(which is 60º), not p3 degrees. When
you do calculus, keep your calculator in
radian mode.
y
y cos x
y sin x
y tan x
x
x – —2 3 – – –2 0 –2 —2 3
– –– 0 –2 —2 3 2 – – –2 0 –2 —2 3 2
2
Function: y
Function: y sin x
tan
Function: y cos x
Domain: x
Domain: – x Domain: – x
–2 , —2 3 , . . .
Range: –1 ≤y ≤1
Period: 2
Range: –1 ≤y ≤1
Period: 2
Range: – y
Period:
(a) (b) (c)
y
y sec x
y
y
y csc x
y
y
y cot x
x
– 3 —2
– –– 2
1
0 –2 —2 3
x
1
– – –2 0 –2 —2 3 2
x
– – –2 0 –2 —2 3 2
1
x
Domain: x –2 , —2 3 , . . .
Domain: x 0, , 2, . . .
Domain: x 0, , 2, . . .
Range: y ≤ –1 and y 1
Range: y ≤ –1 and y 1
Range: – y
Period: 2 Period: 2 Period:
(d)
(e)
(f)
≤
Figure 1.42 Graphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, and
(f) cotangent functions using radian measure.
≤
Periods of Trigonometric
Functions
Period p: tan (x + p) = tan x
cot (x + p) = cot x
Period 2p: sin (x + 2p) = sin x
cos (x + 2p) = cos x
sec (x + 2p) = sec x
csc (x + 2p) = csc x
Periodicity
When an angle of measure u and an angle of measure u 2p are in standard position,
their terminal rays coincide. The two angles therefore have the same trigonometric function
values:
cos (u 2p) cos u sin (u 2p) sin u tan (u 2p) tan u
(1)
sec (u 2p) sec u csc (u 2p) csc u cot (u 2p) cot u
Similarly, cos (u 2p) cos u, sin (u 2p) sin u, and so on.
We see the values of the trigonometric functions repeat at regular intervals. We
describe this behavior by saying that the six basic trigonometric functions are periodic.
DEFINITION
Periodic Function, Period
A function f (x) is periodic if there is a positive number p such that f (x p) f (x)
for every value of x. The smallest such value of p is the period of f.
As we can see in Figure 1.42, the functions cos x, sin x, sec x, and csc x are periodic
with period 2p. The functions tan x and cot x are periodic with period p.
Even and Odd Trigonometric Functions
The graphs in Figure 1.42 suggest that cos x and sec x are even functions because their graphs
are symmetric about the y-axis. The other four basic trigonometric functions are odd.