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[ Pobierz całość w formacie PDF ]Gordon.3896.06.pgs 4/28/03 1:33 PM Page 429
6
Introduction to
Trigonometry
6.1
The Tangent of an Angle
Historically, trigonometry was developed to solve problems involving right trian-
gles. Thus, for the right triangle shown in Figure 6.1, if we know any two of the three
sides
a
,
b
, and
c
, we can easily find the third side with the Pythagorean theorem
c
2
a
2
b
2
,
where
c
is the length of the hypotenuse. But, there is no simple way to find the two
unknown angles. To find them, we need to use trigonometry.
Similarly, if only one of the three sides and one angle are known, we can easily
find the other angle (the two nonright angles must sum to because they are
complementary angles
). However, without trigonometry, there is no simple way to
find the lengths of the other two sides.
The basic idea behind trigonometry is a fundamental geometric fact about
right triangles. The two right triangles shown in Figure 6.2 share the angle (lower-
case Greek letter theta). Therefore the remaining angle in both triangles must be the
same. We denote it (the lowercase Greek letter phi). Because all three angles in
both triangles are the same, the triangles are
similar
(see Appendix A4). As a conse-
quence, once an angle (other than the right angle) has been specified in a right tri-
angle, that triangle is similar to every other right triangle having the same angle
In the smaller triangle shown in Figure 6.2, from the point of view of the
angle
c
b
90°
a
u
FIGURE 6.1
f
u
u.
φ
u,
there is an
adjacent side,
denoted by
a
1
;
an
opposite side,
denoted by
b
1
;
c
2
b
2
and the
hypotenuse,
denoted by
c
1
.
In the larger triangle, also from the point of
φ
view of the angle
u,
the
adjacent side
is
a
2
,
the
opposite side
is
b
2
,
and the
hypotenuse
b
1
c
1
is
c
2
.
The triangles are similar, so that their corresponding sides are proportional, and
θ
a
1
a
1
a
2
b
1
b
2
c
1
c
2
.
a
2
FIGURE 6.2
Equivalently, once an angle has been specified, the ratio of corresponding sides of
these right triangles will be the same. In particular, among several other compara-
ble ratios,
u
a
1
b
1
a
2
b
2
a
1
c
1
a
2
c
2
,
b
1
c
1
b
2
c
2
.
,
and
429
Gordon.3896.06.pgs 4/28/03 1:33 PM Page 430
430
CHAPTER 6
Introduction to Trigonometry
That is, each of these ratios depends solely on the angle not on the dimensions
of the triangle. It is these ratios, and their dependence on the angle that form
the basis of trigonometry. In this section, we begin by examining one of these
three ratios.
u,
u,
The Tangent of an Angle
Suppose that your math instructor has assigned you the task of calculating the
height of a tall flagpole in the middle of campus. The direct approach would be to
climb to the top, release a string until the bottom reaches the ground, and then
measure the length of string. Obviously, this method presents some practical diffi-
culties, and you would likely try to come up with some less physical approach.
Assume that, when you go out to the flagpole, you notice that the pole is cast-
ing a 66-foot-long shadow. How can you use this piece of extra information to de-
termine the height of the pole? Suppose that you enlist the aid of a friend Ron, who
is exactly six feet tall. Have him stand in the shadow cast by the pole so that the tip
of his shadow falls exactly on the same spot
A
as the tip of the shadow of the flag-
pole, as illustrated in Figure 6.3. Also, suppose that the length of his shadow is
or feet. The two triangles
ABC
and
ADE
are similar because the angles are the
same, and so the corresponding sides are proportional. Therefore
8
4
,
8.25,
Ron’s height
height of pole
length of pole’s shadow
length of his shadow
height of pole
66
6
8.25
.
C
?
E
6
4
8
A
D
B
FIGURE 6.3
66
Multiplying both sides by 66 yields
6
2
8.25
1
66
Height of the pole
48 feet.
Is this result correct? You can check it with the help of another friend, Sue, who
is five feet tall. Have her stand so that the tip of her shadow matches the end of the
pole’s shadow. Suppose that the length of her shadow is or feet, which
leads to right triangle
AFG
that is similar to the previous two, as illustrated in Fig-
ure 6.4. Because the corresponding sides are proportional, we get
6
8
,
6.875,
Gordon.3896.06.pgs 4/28/03 1:33 PM Page 431
6.1
The Tangent of an Angle
431
C
?
E
G
6
5
A
6
8
B
F
D
FIGURE 6.4
66
Sue’s height
height of pole
length of pole’s shadow
length of her shadow
height of pole
66
5
6.875
.
Again, we find that
5
2
6.875
1
66
Height of the pole
48 feet.
Let’s look at this situation from a slightly more sophisticated point of view. In
each of the three right triangles shown in Figure 6.5, the various lengths are differ-
ent but the angles in corresponding positions are all the same, so all three triangles
are similar. The angle (which is the same as angle
CAB
, angle
EAD
, and angle
GAF
) is called the
angle of inclination
. Using a protractor, we measure this angle
and find that is about In fact, in
any
right triangle where the angle of incli-
nation is the ratio of the vertical height (the opposite side) to the horizontal
distance or width (the adjacent side) will always be the same; in this case,
Height
Width
u
u
36°.
36°,
6
8.25
5
6
8
0.727.
C
E
G
FIGURE 6.5
θ
A
F
D
B
Of course, if the angle has a different value—say, —the configura-
tion of height and width is different and their ratio therefore is different. The ratio
of height to width, or opposite side to adjacent side, in a right triangle depends
only on the size of the angle
u
u
40°
u,
so this ratio is a function of the angle. We call this
Gordon.3896.06.pgs 4/28/03 1:33 PM Page 432
432
CHAPTER 6
Introduction to Trigonometry
function the
tangent of the angle,
the
tangent ratio,
or the
tangent function
and
write it as
opposite
adjacent
height
width
tan u
.
Use your calculator, in
Degree
mode, to verify that (Note that
the values of the tangent function, as well as the other trigonometric functions that
we discuss in Section 6.2, typically are irrational numbers, but we usually give the
values to three or four decimal places.)
Because we are concerned exclusively with right triangles here, the angle
must be between and and so for now the domain of the tangent function
consists of all angles (Later we show how we can extend it to a larg-
er domain.) Also, we can have a right triangle in any possible orientation, as shown
in Figure 6.6, so the words
height
and
width
may not be appropriate. Instead, we
typically think of the tangent ratio for an angle
tan 36°
0.7265.
u
0°
90°,
0°
u
90°.
u
as follows.
opposite
adjacent
tan u
Adjacent
Opposite
θ
Hypotenuse
FIGURE 6.6
From the point of view of the other angle in the triangle, the opposite and
adjacent sides are reversed, as depicted in Figure 6.7. Note also that the angles
f
u
and
f
are complementary angles.
The Tangent of Some “Special” Angles
Recall from geometry that in any –– right triangle the two sides flanking
the hypotenuse are equal and, by the Pythagorean theorem,
45°
45°
90°
c
2
a
2
a
2
2
a
2
,
so
c
1
2
a
.
That is, the hypotenuse must be
1
2
times the length of either side
,
as il-
lustrated in Figure 6.8. In this triangle with angle
u
45°
and sides
a
,
a
, and
1
2
a
,
45°
φ
√2
a
Hypotenuse
a
Adjacent
θ
45°
FIGURE 6.7
FIGURE 6.8
Opposite
a
Gordon.3896.06.pgs 4/28/03 1:33 PM Page 433
6.1
The Tangent of an Angle
433
opposite
adjacent
a
a
tan 45°
1.
You can easily verify that
tan 45°
1
on your calculator. (Be sure that your calcu-
lator is set in
Degree
mode.)
Similarly, recall from geometry that in any –– right triangle, the side
opposite the angle is one-half the hypotenuse, or, equivalently, the hypotenuse
is twice the side opposite the angle. In such a triangle, suppose that the side op-
posite the angle has length
a
so that the hypotenuse has length as shown in
Figure 6.9. We find the length of the third side from the Pythagorean theorem. Be-
cause
30°
60°
90°
30°
30°
30°
2
a
,
a
2
b
2
c
2
,
b
2
c
2
a
2
,
we have
so
b
2
2
a
2
4
a
2
a
2
3
a
2
1
2
a
2
so that
b
2
3
a
.
60°
2
a
a
30°
b
= √3
a
FIGURE 6.9
Consequently, for an angle of
30°,
the ratio of the opposite side to the adjacent side is
a
1
1
1
tan 30°
3
a
3
0.577.
Alternatively, using a calculator, we find
Similarly, to find t
h
e tangent of
tan 30°
0.577.
60°,
we see from Figure 6.9 that the side oppo-
site the
60°
angle is
1
3
a
and the side adjacent to it is
a
, so that
1
3
a
a
tan 60°
1
3
1.732,
which you can also check on your calculator.
For any angle
u
between
0°
and
90°,
you can use a calculator to obtain the cor-
responding value for
tan u.
For instance, to three decimal place accuracy,
tan 10°
0.176,
tan 20°
0.364,
tan 50°
1.192,
tan 80°
5.671.
Note that as increases toward the value of also increases; that is, the
tangent is an increasing function of at least between and Does that make
sense? Imagine walking toward the 556-foot-high Washington Monument while
keeping your eye fixed on the top of the monument, as illustrated in Figure 6.10.
u
90°,
tan u
u,
0°
90°.
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