Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
10.3 Properties of the trigonometric functions
We now explore the consequences of these definitions:
Values of sine and cosine
• The radius of the circle is 1. This means that the x coordinate cannot be larger than 1 or
smaller than -1. Same holds for the y coordinate. Thus the functions sin(t) and cos(t) are
always swinging between -1 and 1. (−1 ≤ sin(t) ≤ 1 and −1 ≤ cos(t) ≤ 1 for all t). The peak
(maximum) value of each function is 1, the minimum is -1, and the average value is 0.
• When the radius vector points along the x axis, the angle is t = 0 and we have y = 0, x = 1.
This means that cos(0) = 1, sin(0) = 0.
• When the radius vector points up the y axis, the angle is π/2 (corresponding to one quarter
of a complete revolution), and here x = 0, y = 1 so that cos(π/2) = 0, sin(π/2) = 1.
• Using simple geometry, we can also determine the lengths of all sides, and hence the ratios of
the sides in a few particularly simple triangles, namely triangles (in which all angles are 60 ◦ ),
and right triangles with two equal angles of 45 ◦ . These types of calculations (omitted here)
lead to some easily determined values for the sine and cosine of such special angles. These
values are shown in the table below.
degrees radians sin(t) cos(t) tan(t)
0 0 0 1 0
π
30
6
π
45
4
Connection between sine and cosine
1
√2
2
√2
3
2
√
3 1√
√2
3
2
2
1
√
π
1
60
3
2 3
π
90
2
1 0 ∞
• The two functions, sine and cosine depict the same underlying motion, viewed from two
perspectives: cos(t) represents the projection of the circularly moving point onto the x axis,
while sin(t) is the projection of that point onto the y axis. In this sense, the functions are a
pair of twins, and we can expect many relationships to hold between them.
• The cosine has its largest value at the beginning of the cycle, when t = 0 (since cos(0) = 1),
while the other the sine its peak value a little later, (sin(π/2) = 1). Throughout their circular
race, the sine function is π/2 radians ahead of the cosine i.e.
cos(t) = sin(t + π 2 ).
• The point (x, y) is on a circle of radius 1, and, thus, its coordinates satisfy
x 2 + y 2 = 1
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