Game Development Reference

In-Depth Information

triangle based on the angle we

re interested in; in this case
A
.The

vertical side of the triangle is
opposite
(opp) angle
A
,whilethe

horizontal side is
adjacent
(adj) to it.

The aforementioned trig functions work with these sides as

follows:

'

90
°

The sine of an angle is equal to the opposite side

'

s length

A

divided by the hypotenuse

'

s length: sin
A = opp/hyp

Adjacent

The cosine of an angle is equal to the adjacent side

'

s length

Figure 11.3
The three

sides of a right triangle,

related to angle A.

s length: cos
A = adj/hyp

The tangent of an angle is equal to the opposite side divided by

the adjacent site: tan
A = opp/adj

As you can see, these functions are very helpful if you only know

a little bit of information about a triangle and need to determine the

other components. Let

divided by the hypotenuse

'

s look at a few examples. In
Fig. 11.4
,we

know the value of angle
A
is 50

'

°

(and by extension, the other

missing angle would then be 40

°

). We also know the length of the

hypotenuse is 30.

To find the lengths of the other two sides, we rewrite the sine

and cosine equations as follows:

90
°

A
=
50
°

adj

=

cos
A

×

hyp
,or
adj

=
ð

cos 50

Þ
×

30

Adjacent

opp

=

sin
A

×

hyp
,or
opp

=
ð

sin 50

Þ
×

30

Figure 11.4
Using the

information about one

angle and one side,

we can use the trig

functions to find the

values of the other

two sides.

If you used a calculator with the trig functions on it, you would

quickly determine that the value of the adjacent side is ~19.3 and

the value of the opposite side is ~23.

In
Fig. 11.5
, we can see that we now know one angle (45

°

)and

the length of the side opposite that angle (20

).

Once again, we simply manipulate the equations to determine

the other two sides, this time using tangent instead of cosine, since

cosine has nothing to do with the opposite side:

°

hyp

=

opp
/ sin
A
,or
hyp

=

20/

sin 45

Þ

ð

adj

=

opp
/ tan
A
,or
adj

=

20/

ð

tan 45

Þ

Using a calculator, this would reveal the hypotenuse to have a

length of ~28.3 and the adjacent side to also be 20.

Now let

90
°

A
=
45
°

Adjacent

s look at an example (
Fig. 11.6
) with a triangle where

we know the lengths of the two shorter sides, but no angles and no

hypotenuse.

Since we know the opposite and adjacent sides, the obvious

choice would be to use the tangent equation to determine the value

of angle
A
(and flipping the two sides to find out the value of
B
):

'

Figure 11.5
A triangle

where we know one

angle and one side.

tan
A

=

15/20

tan
B

=

20/15

re stuck. We want the values of
A
and
B
,not

the tangent of
A
and
B
. Luckily, there is a way to reverse each trig

However, now we

'