The Sine and Cosine Functions
The Graphs of Sine and Cosine
In the last lecture, we defined the quantities sinθand cos θfor all angles θ. Today we explore the sine and
cosine functions, their properties, their derivatives, and variations on those two functions.
By now, you should have memorized the values of sin θand cos θfor all of the special angles. For the
purposes of completeness, we recreate the tables of these values from the last lecture below:
θcos θsin θ
0 1 0
π
6
√3
2
1
2
π
4
√2
2
√2
2
π
3
1
2
√3
2
π
20 1
2π
3−1
2
√3
2
3π
4-√2
2
√2
2
5π
6-√3
2
1
2
π-1 0
θcos θsin θ
7π
6-√3
2-1
2
5π
4−√2
2−√2
2
4π
3−1
2−√3
2
3π
20 -1
5π
3
1
2−√3
2
7π
4
√2
2−√2
2
11π
6
√3
2−1
2
2π1 0
Clearly, we can plot the function f(x) = sin xand the function g(x) = cos x(replacing θwith x) using
the numerical tables above. We note that √2
2≈0.71, √3
2≈0.87, π≈3.14, π
2≈1.57, π
4≈0.79, π
6≈0.52,
and π
3≈1.05.
The resulting graph of f(x) = sin xlooks like the following: we begin at the origin and with a positive
slope of 1. As xincreases, the slope of the sine function decreases, so that the derivative is 0 when we get
to the point (π
2,1), which is a local maximum for sin x. The function then begins to decrease, first with a
gentle slope, but by the point when the graph crosses the x-axis, at the point (π, 0), the slope of the graph
of the sine function is −1. This point is also an inflection point, so the graph, which up until now was always
concave down, now begins to be concave up. The slope of the graph starts becoming less negative, so that
at the point ( 3π
2,−1) the graph has zero slope. This point is at a local minimum for the sine function, so
the graph now rises again, its slope increasing, until at (2π, 0) its slope is 1 again.
Instead of continuing the graph of f(x) = sin xin both directions, let us now attempt to sketch the graph
of g(x) = cos x. We begin at the point (0,1). Here, the slope of the cosine function will be zero, and the
function will have a local maximum, so the graph will begin to drop, first with a gentle negative slope, until
finally it reaches the point (π
2,0), where its slope is −1. This point is an inflection point, so now the graph
goes from concave down to concave up. The slop e of the graph increases, so that at (π, −1) we have a critical
point of the function, and a local minimum. The graph now starts to rise, and by the point it reaches the
x-axis, which is (3π
2,0), the slope of the graph is 1, and we have another inflection point, this time from
concave up to concave down. The graph continues to rise, but more and more slowly, and then, at (2π, 1),
the slope is zero and we are at a local maximum again.
What about the graphs for xgreater than 2πand for negative x? Here we use the fact that angles whose
arcs have the same endpoint have the same sine and cosine. This means that, every time xchanges by 2π
(the number of radians in a circle), the graphs of sine and cosine repeat themselves. So we sketch in the
graphs of sin xand cos xfor xgreater than 2πand negative xby simply continuing the graph just as it had
started and stopped. The only new point to make here is that (0,0) and (0,2π) are now clearly inflection
points, where the graphs goes from concave up to concave down.
The Properties of Sine and Cosine
Let us now list in algebraic form the properties of the sine and cosine functions:
•Periodicity: Both sin xand cos xhave the property that they repeat themselves every time xincreases
(or decreases) by 2π. Algebraically, we write this property as
sin(x+ 2π) = sin xand cos(x+ 2π) = cos x.
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