Game Development Reference
In-Depth Information
Table 12.5 Generating Tangent and Cotangent Values
Item
Discussion
sin y
cos y
tan y ¼
If the value of the of cos y in this function is 0, then the value of the tangent is undefined.
When you plot tangent values on a Cartesian plane, the resulting curve rises indefinitely as
it approaches a line extending vertically from any point on the x coordinate at which cos y
is 0. To formally state this, you can say that tan u is not defined at any value of 2 þ kp.In
this case, k is any integer value. As you see in Figure 12.18, such values are 2 , 2 , 3 2 , and
5 2 . Given this situation, the period of the tangent values is p.
cos y
sin y
cot y ¼
If the value of the of sin y is 0, then the value of the cosecant is undefined. When you plot
cotangent values on a Cartesian plane, the resulting curve rises indefinitely as it
approaches a line extending vertically from any point on the x coordinate at which sin
y
is
0. A formal way to say this is that csc
y
is not defined at any value of
p þ k
p
. The value
of k is any integer. As you see in Figure 12.19, such values are
p
,2
p
, p
, 2
p
,3
p
, and 0.
Given this situation, the period of the tangent values is
p
.
sec
y ¼
1
cos y
The cosine and secant functions are reciprocals of each other. The value of the cosecant is
undefined when it falls on a vertical line passing through a point on the x axis at which the
cosine value is 0. Given this situation, the period of the secant values is 2
p
.
1
sin y
csc
y ¼
The sine and cosecant functions are reciprocals of each other. The value of the cosecant is
undefined when it falls on a vertical line passing through a point on the x axis at which the
sine value is 0. Given this situation, the period of the cosecant values is 2
p
.
Defining Tangents and Cotangents
Table 12.5 provides you with a summary of how to generate tangent and
cotangent values. To generate a tangent value, you divide sin
by cos
.To
generate a secant value, you use the reciprocal of cos
. A variety of approaches to
arriving at the different values of the trigonometric functions exist. The approach
given in Table 12.5 proves one of the easiest to follow.
Plotting Tangent Values
When you plot tangent values on the Cartesian plane, you cannot define them at
certain points. These points are those that occur as 0 when you plot cosine values.
Such values are 2 , 2 ,
2 . You can see why this occurs if you consider
the shapes of the periodic waves that characterize the plotting of the cosine of
3
2 ,
3
y
(see Figure 12.17 earlier in this chapter). Whenever the value of the cosine
reaches 0, then the value of the tangent is undefined.
It is undefined because, as you plot tangent values on a Cartesian plane, the
resulting curve rises or falls indefinitely as it approaches lines extending vertically
 
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