TRIGONOMETRY
FUNDAMENTALS
Up to this point we have been learning the language of
mathematics and laying the foundation for trigonometry. Along the way we have
discussed
arithmetic,
relationships
and operations of numbers;
algebra,
a type of generalized arithmetic involving both constant numbers
and variables; and
geometry,
the study of
forms and relationships of plane figures.
Trigonometry
is a branch of mathematics that includes arithmetic,
algebra, and geometry, but is primarily concerned with the relationships of
lines and angles in triangles. It is the basis of measurements used in
surveying, engineering, shop mechanics, geodesy, and astronomy. The word
trigonometry
comes from the Greek words
trigonon,
a triangle, and
metron,
to measure.
Trigon
metron
measure
We begin our exploration of trigonometry by focusing strictly on
right triangles, which form the basis for all trigonometric calculations.
Recall that a right triangle is any triangle with a 90° angle. The standard
right triangle is illustrated in Figure 20.1. Note that the three angles are
labeled by the capital letters
A
,
B
, and
C
. Sometimes the three-letter
convention is used to name the angles, in which case the vertex is the center
letter, as in
∠
ABC
also referred to as
∠
B
. When the three vertices are
used to name a right triangle, the vertex containing the right angle is
sometimes underlined as in triangle
ACB
.
By convention,
∠
C
is always the 90° angle.
The word trigonometry is often informally shortened
to “trig.”
F
IGURE
20.1
Conventional right triangle
designations
The sides of triangles are identified with lowercase letters.
Side
a
, for example, is the side
opposite acute
∠
A
and side
b
is the side opposite acute
∠
B
. The side labeled
c
is always opposite to the 90°
angle and is therefore always the hypotenuse. Of course, the term hypotenuse is
already familiar from our study of the Pythagorean theorem and it is easily
recognized as the longest side in a right triangle. Sides
a
and
b
are the
legs
of the right triangle.
“Opposite” and “adjacent” are
always relative to the specified angle.
In reference to sides
a
and
b
, note the importance of
understanding what is meant by “opposite side” and “adjacent side.” These are
relative terms, which change depending on which acute angle of a right triangle
is under consideration. In Figure 20.1, side
a
is adjacent to
∠
B
while
side
b
is opposite to
∠
B
. Likewise, side
b
is adjacent to
∠
A
while side
a
is opposite to
∠
A
. The reference chosen depends on
the problem to be solved. The hypotenuse does not change and is always opposite
the right angle.
Greek letters such as
,
, and
(alpha, beta, and theta) are commonly used to
denote the acute angles in right-triangles.
Table 20.1 provides a summary of terms used in trigonometry.
Trigonometric
Functions
The underlying premise for trigonometry is that for
any
right triangle, the ratios of the
lengths of any two sides relative to a given acute angle is the same. In other words,
regardless of the size of a triangle, the proportions of the sides remain constant
if the angles remain the same. Taking two sides at a time, we can readily see
that six such ratios are possible for a right triangle. Each ratio is a
specific trigonometric function.
The six trigonometric functions are:
sine
(abbreviated
sin
);
cosine
(abbreviated
cos
);
tangent
(abbreviated
tan
);
cotangent
(abbreviated
cot
);
secant
(abbreviated
sec
) and
cosecant
(abbreviated
csc.
)
The functions, named in reference to a standard right triangle,
are defined and shown in Figures 20.2 through 20.13. As before, we follow
convention and letter the right angle
C
and
its opposite side, the hypotenuse, as
c
.
The legs, or the shorter sides of the triangle, are labeled
a
and
b
and are opposite the acute angles
A
and
B,
respectively.
SOHCAHTOA
(soh cah toa) is a made-up word
to help remember the primary trigonometric relationships: ''
S
ine is
O
pposite over
H
ypotenuse,
C
osine is
A
djacent over
H
ypotenuse,
T
angent is
O
pposite over
A
djacent.''
These six functions are the basis for all work in trigonometry
and should be learned thoroughly to facilitate the solving of shop problems.
A “3-4-5” triangle is a triangle with its sides in
the ratio of 3 to 4 to 5.
EXAMPLE 20.1A:
Finding Values for
Trigonometric Functions
Consider the 3-4-5 triangle shown in Figure 20.14. Find the
sine, cosine, tangent, cotangent, secant and cosecant of the acute angles.
FIGURE 20.14
Notice that 32 + 42 = 52,
as the Pythagorean theorem claims.
Solution:
The two acute angles are 36.9°
and 53.1°.
EXAMPLE 20.1B:
Finding Values for
Trigonometric Functions
Consider the 3-4-5 triangle shown in Figure 20.15. While the
angles are the same as in the previous example, the lengths of the sides are
different. Find the sine, cosine, tangent, cotangent, secant, and cosecant of
the acute angles.
FIGURE 20.15
Solution
:
Again, the two acute angles are
36.9° and 53.1°.
Notice that although the sizes of the triangles in Examples 20.1A
and 20.1B changed, the
proportions
of corresponding sides were the same. Hence, the angles were the
same, and so were the values of the trigonometric functions for each acute
angle. This fact is the key to how trigonometry works.
Because the trigonometric functions are ratios of
side lengths whose units are the same, the values of the functions are
unitless. For example,
Since the angles in Examples 20.1A and 20.1B are given, we could
also find the values of the trigonometric functions directly from a scientific
calculator without actually using the ratios of the sides. Verify the answers
using a scientific calculator.
Several important relationships between certain pairs of
trigonometric functions are useful for solving problems. These relationships,
known as the
complementary
and
reciprocal relationships
, are discussed next.
Complementary
Trigonometric Relationships
From our knowledge of geometry, we know that the sum of the
measures of the interior angles of a triangle must add to 180. The measures of
the acute angles of right triangle, therefore, must add to 90. Referring to the
standard triangle shown in Figure 20.1,
A
and
B
must always be
complements. Accordingly,
m
∠
A
= 90 –
m
∠
B
and we can formulate the
complementary trigonometric relationships given in Table 20.2.
Another way to look at the complementary trigonometric
relationships is to observe that sin
A
is
identical to cos
B
. This is due to the fact that
for both functions the ratios involve the same sides. Likewise, tan
A
= cot
B
. These relationships are known
as the trigonometric
cofunction identities
and are summarized in Table 20.3 in reference to Figure 20.1.
TABLE 20.2:
Complementary Trigonometric Identities
TABLE 20.3:
Trigonometric
Cofunction Identities
EXAMPLE 20.2:
Trigonometric Cofunction
Identities
Fill in the blanks in the following statements and verify
results with a scientific calculator:
sin 23° = cos____ cos 39° =
sin____ tan 10° = cot____
EXAMPLE 20.2:
Trigonometric Cofunction
Identities (Continued)
Reciprocal
Trigonometric Relationships
Recall that the reciprocal of a fraction is obtained by
interchanging the numerator and the denominator. For instance, the reciprocal
of is . The reciprocal of a whole number such as 2 is since the denominator of
a whole number is simply “1.” Zero has no reciprocal, since inverts to , which
is undefined. The definitions of the six trigonometric functions reveal that
each function is the reciprocal of one of the other functions. These
relationships are called the trigonometric
reciprocal identities
and are summarized in Table 20.4.
TABLE 20.4:
Reciprocal
Trigonometric Identities
Because of the reciprocal relationships, many scientific
calculators only have keys for sine, cosine, and tangent. The cosecant, secant,
and cotangent functions are readily found by using the reciprocal function
found on most scientific calculators.
On many calculators the reciprocal key is
identified as 1/x or x
-
1
.
EXAMPLE 20.3:
Reciprocal Trigonometric
Identities
Fill in the blanks in the following statements and verify
results with a scientific calculator:
Copyright © 2006 Industrial
Pres s Inc.