College Trigonometry, 2011a

10.5 Graphs of the Trigonometric Functions 791
corners or cusps. As we shall see, the graphs of both f(t) = cos(t) andg(t) =sin(t) meander nicely
and don’t cause any trouble. We summarize these facts in the following theorem.
Theorem 10.22. Properties of the Cosine and Sine Functions
• The function f(x) = cos(x)
• The function g(x) =sin(x)
– has domain (−∞, ∞) – has domain (−∞, ∞)
–hasrange[−1, 1] – has range [−1, 1]
– is continuous and smooth – is continuous and smooth
–iseven
–isodd
– has period 2π – has period 2π
In the chart above, we followed the convention established in Section 1.6 and used x as the independent
variable and y as the dependent variable. 5 This allows us to turn our attention to graphing
the cosine and sine functions in the Cartesian Plane. To graph y = cos(x), we make a table as we
did in Section 1.6 using some of the ‘common values’ of x in the interval [0, 2π]. This generates a
portion of the cosine graph, which we call the ‘fundamental cycle’ ofy = cos(x).
x cos(x) (x, cos(x))
0 1 (0, 1)
( √ )
π
4 , 2
π
√
2
4 2
π
2
0
3π
4
−
√
2
2
2
( π
( 2 , 0)
√ )
3π
4 , − 2
2
π −1 (π, −1)
( √ )
5π
4 , − 2
√
5π
4
− 2
2
3π
2
0
7π
4
√
2
2
2
( 3π
( 2 , 0)
√ )
7π
4 , 2
2
2π 1 (2π, 1)
1
−1
y
π
4
π
2
3π
4
The ‘fundamental cycle’ of y = cos(x).
π
5π
4
3π
2
7π
4
2π
x
A few things about the graph above are worth mentioning. First, this graph represents only part
of the graph of y = cos(x). To get the entire graph, we imagine ‘copying and pasting’ this graph
end to end infinitely in both directions (left and right) on the x-axis. Secondly, the vertical scale
here has been greatly exaggerated for clarity and aesthetics. Below is an accurate-to-scale graph of
y = cos(x) showing several cycles with the ‘fundamental cycle’ plotted thicker than the others. The
5 The use of x and y in this context is not to be confused with the x- andy-coordinates of points on the Unit Circle
which define cosine and sine. Using the term ‘trigonometric function’ as opposed to ‘circular function’ can help with
that, but one could then ask, “Hey, where’s the triangle?”