Game Development Reference
In-Depth Information
Note
When working with the values shown in Table 12.3, it is common to manipulate the resulting
quotients so that radicals do not appear in the denominators. To eliminate the radical, you
multiply by 1 using the value given by the square root. Here are a few examples:
p
p ¼
3
2
3
p
hypotenuse
adjacent
¼
2
p
sec 30
8 ¼
3
p
p
¼
3
p
3
adjacent
opposite
¼
1
p
cot 60
8 ¼
3
Rotation
The trigonometric ratios all translate into functions that can generate distinct
patterns when applied to a Cartesian plane. As you increase the values you in-
troduce to the functions, the patterns change in restricted, predictable ways. The
most ready way to generate such patterns involves translating the three sides of
the standard triangle (opposite, adjacent, and hypotenuse) so that you can
understand them in relation to the coordinates you plot on the Cartesian plane
(
x
,
y
). Work earlier in this chapter anticipates this activity. Table 12.4 shows you
how the sides of a triangle relate to the values you generated using values typical
of your work with the Cartesian plane. In each instance, you work with a stan-
dard triangle.
As Figure 12.12 illustrates, if you use combinations of negative and positive
values for
x
and
y,
you rotate the triangle around the origin of the plane through
all four quadrants. The values
x
and
y
correspond to the values of
x
and
y
on the
axes of the plane if the vertex of angle
resides at (0,0).
Table 12.4
Trigonometric Ratios
Item
Ratio
Mnemonic
opposite
hypotenuse
y
r
Sine
sin
y ¼
sin
y ¼
adjacent
hypotenuse
x
r
Cosine
cos
y ¼
cos
y ¼
Tangent
opposite
adjacent
y
x
tan
y ¼
tan
y ¼
Cotangent
adjacent
opposite
x
y
cot
y ¼
cot
y ¼
Secant
hypotenuse
adjacent
r
x
sec
y ¼
sec
y ¼
Cosecant
hypotenuse
opposite
csc
y ¼
csc
y ¼
r
y
