5.1 Trigonometric Functions of Acute Angles

234 Pages·2006·6.15 MB·English

by  Judith A. Beecher

Summer 2012
Advanced Algebra & Trigonometry Course Content

“Algebra and Trigonometry” (Third Edition)
Beecher, Penna, Bittinger
Addision Wesley (February 2, 2007)
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The Trigonometric
5
Functions
5.1 Trigonometric Functions of Acute Angles
5.2 Applications of RightTriangles
5.3 Trigonometric Functions of Any Angle
5.4 Radians,Arc Length,and Angular Speed
5.5 Circular Functions:Graphs and Properties
5.6 Graphs of Transformed Sine and
Cosine Functions
SUMMARYANDREVIEW
TEST
A P P L I C A T I O N
I
n Comiskey Park,the home ofthe Chicago White
Sox baseball team,the angle ofdepression from
a seat in the last row ofthe upper deck directly
behind the batter to home plate is 29.9(cid:1),and the angle
ofdepression to the pitcher’s mound is 24.2(cid:1).Find
(a) the viewing distance to home plate and (b) the
viewing distance to the pitcher’s mound.
This problem appears asExample 7 in Section 5.2.
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Chapter 5 • The Trigonometric Functions
25..11
Determine the six trigonometric ratios for a given acute angle ofa
right triangle.
Determine the trigonometric function values of 30(cid:1),45(cid:1),and 60(cid:1).
Using a calculator,find function values for any acute angle,and given a
function value ofan acute angle,find the angle.
TrigPoonlyonmoemtiraicl
Given the function values ofan acute angle,find the function values of
FuFnucntciotinosn as nodf its complement.
AcuMteo Adneglilnegs
The Trigonometric Ratios
We begin our study of trigonometry by considering right triangles and
acute angles measured in degrees.An acute angle is an angle with mea-
sure greater than 0(cid:1) and less than 90(cid:1). Greek letters such as (cid:6)(alpha),
(cid:5)(beta), (cid:4)(gamma), (cid:2)(theta), and (cid:3)(phi) are often used to denote an
angle. Consider a right triangle with one of its acute angles labeled (cid:2).
The side opposite the right angle is called the hypotenuse. The other
sides of the triangle are referenced by their position relative to the acute
angle (cid:2).One side is opposite (cid:2)and one is adjacent to (cid:2).
Hypotenuse
Side opposite u
u
Side adjacent to u
The lengths of the sides of the triangle are used to define the six
trigonometric ratios:
sine (sin), cosecant (csc),
cosine (cos), secant (sec),
tangent (tan), cotangent (cot).
Hypotenuse Opposite u The sine of(cid:1)is the lengthof the side opposite (cid:2)divided by the length
of the hypotenuse (see Fig.1):
u
Adjacent to u length of side opposite (cid:2)
sin (cid:2)(cid:1) .
Figure 1 length of hypotenuse
The ratio depends on the measure of angle (cid:2)and thus is a function of (cid:2).
The notation sin (cid:2)actually means sin (cid:1)(cid:2)(cid:2),where sin,or sine,is the name
of the function.
Hypotenuse Opposite u The cosine of (cid:1)is the length of the side adjacent to (cid:2)divided by the
lengthofthe hypotenuse (see Fig.2):
u
Adjacent to u length of side adjacent to (cid:2)
cos (cid:2)(cid:1) .
Figure 2 length of hypotenuse
The six trigonometric ratios, or trigonometric functions, are defined as
follows.
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Section 5.1 • Trigonometric Functions of Acute Angles
Trigonometric Function Values of an Acute Angle (cid:1)
Let (cid:2)be an acute angle ofa right triangle.Then the six trigonometric
functions of(cid:2)are as follows:
side opposite (cid:2) hypotenuse
sin (cid:2)(cid:1) , csc (cid:2)(cid:1) ,
Hypotenuse hypotenuse side opposite (cid:2)
Opposite u
side adjacent to (cid:2) hypotenuse
u cos (cid:2)(cid:1) , sec (cid:2)(cid:1) ,
Adjacent to u hypotenuse side adjacent to (cid:2)
side opposite (cid:2) side adjacent to (cid:2)
tan (cid:2)(cid:1) , cot (cid:2)(cid:1) .
side adjacent to (cid:2) side opposite (cid:2)
EXAMPLE 1 In the right triangle shown at left,find the six trigonomet-
ric function values of (a)(cid:2)and (b)(cid:6).
a
12 13 Solution We use the definitions.
opp 12 hyp 13
u a) sin (cid:2)(cid:1) (cid:1) , csc (cid:2)(cid:1) (cid:1) ,
hyp 13 opp 12
5
The references to
adj 5 hyp 13
cos (cid:2)(cid:1) (cid:1) , sec (cid:2)(cid:1) (cid:1) , opposite,adjacent,and
hyp 13 adj 5 hypotenuse are relative
to (cid:1).
opp 12 adj 5
tan (cid:2)(cid:1) (cid:1) , cot (cid:2)(cid:1) (cid:1)
adj 5 opp 12
opp 5 hyp 13
b) sin (cid:6)(cid:1) (cid:1) , csc (cid:6)(cid:1) (cid:1) ,
hyp 13 opp 5
The references to
adj 12 hyp 13
cos (cid:6)(cid:1) (cid:1) , sec (cid:6)(cid:1) (cid:1) , opposite,adjacent,and
hyp 13 adj 12 hypotenuse are relative
to (cid:2).
opp 5 adj 12
tan (cid:6)(cid:1) (cid:1) , cot (cid:6)(cid:1) (cid:1)
adj 12 opp 5
In Example 1(a), we note that the value of sin (cid:2), 12, is the reciprocal
13
of 13,the value of csc (cid:2).Likewise,we see the same reciprocal relationship
12
between the values of cos (cid:2)and sec (cid:2)and between the values of tan (cid:2)and
cot (cid:2). For any angle, the cosecant, secant, and cotangent values are the
reciprocals of the sine, cosine, and tangent function values, respectively.
Reciprocal Functions
1 1 1
csc (cid:2)(cid:1) , sec (cid:2)(cid:1) , cot (cid:2)(cid:1)
sin (cid:2) cos (cid:2) tan (cid:2)
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Chapter 5 • The Trigonometric Functions
If we know the values of the sine, cosine, and tangent functions of an
angle, we can use these reciprocal relationships to find the values of the
cosecant,secant,and cotangent functions of that angle.
Study Tip EXAMPLE 2 Given that sin (cid:3)(cid:1) 4,cos (cid:3)(cid:1) 3,and tan (cid:3)(cid:1) 4,find csc (cid:3),
5 5 3
sec (cid:3),and cot (cid:3).
Success on the nextexam can
be planned.Include study time Solution Using the reciprocal relationships,we have
(even if only 30minutes a day) in
1 1 5 1 1 5
your daily schedule and committo csc (cid:3)(cid:1) (cid:1) (cid:1) , sec (cid:3)(cid:1) (cid:1) (cid:1) ,
sin (cid:3) 4 4 cos (cid:3) 3 3
making this time a priority.Choose
a time when you are mostalert 5 5
and a setting in which you can
and
concentrate.You will be surprised
how much more you can learn and 1 1 3
cot (cid:3)(cid:1) (cid:1) (cid:1) .
retain if study time is included tan (cid:3) 4 4
each day rather than in one long
3
session before the exam.
Triangles are said to be similar if their corresponding angles have
the same measure.In similar triangles,the lengths of corresponding sides
are in the same ratio. The right triangles shown below are similar. Note
that the corresponding angles are equal and the length of each side of the
second triangle is four times the length of the corresponding side of the
first triangle.
a
20
12
a 5
3
b b
4 16
Let’s observe the sine,cosine,and tangent values of (cid:5)in each triangle.
Can we expect corresponding function values to be the same?
FIRSTTRIANGLE SECONDTRIANGLE
3 12 3
sin (cid:5)(cid:1) sin (cid:5)(cid:1) (cid:1)
5 20 5
4 16 4
cos (cid:5)(cid:1) cos (cid:5)(cid:1) (cid:1)
5 20 5
3 12 3
tan (cid:5)(cid:1) tan (cid:5)(cid:1) (cid:1)
4 16 4
For the two triangles, the corresponding values of sin (cid:5), cos (cid:5), and
tan (cid:5)are the same. The lengths of the sides are proportional—thus the
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Section 5.1 • Trigonometric Functions of Acute Angles
ratios are the same. This must be the case because in order for the sine,
cosine, and tangent to be functions, there must be only one output (the
ratio) for each input (the angle (cid:5)).
The trigonometric function values of (cid:2)depend only on the measure
ofthe angle,noton the size ofthe triangle.
The Six Functions Related
We can find the other five trigonometric function values of an acute angle
when one ofthe function-value ratios is known.
EXAMPLE 3 If sin (cid:5)(cid:1) 6 and (cid:5)is an acute angle, find the other five
7
trigonometric function values of(cid:5).
Solution We know from the definition of the sine function that the ratio
6 opp
is .
7 hyp
Using this information, let’s consider a right triangle in which the
pythagorean theorem
hypotenuse has length 7 and the side opposite (cid:5)has length 6. To find the
review section 1.1. length ofthe side adjacent to (cid:5),we recall the Pythagorean theorem:
a2 (cid:8) b2 (cid:1) c2
a2 (cid:8) 62 (cid:1) 72
7 6 a2 (cid:8) 36 (cid:1) 49
a2 (cid:1) 49 (cid:7) 36 (cid:1) 13
b a (cid:1) (cid:3)13.
a
We now use the lengths ofthe three sides to find the other five ratios:
6 7
sin (cid:5)(cid:1) , csc (cid:5)(cid:1) ,
7 6
(cid:3)13 7 7(cid:3)13
7 6 cos (cid:5)(cid:1) , sec (cid:5)(cid:1) , or ,
7 (cid:3)13 13
6 6(cid:3)13 (cid:3)13
b tan (cid:5)(cid:1) , or , cot (cid:5)(cid:1) .
(cid:3)13 13 6
(cid:3)13
30(cid:3) 45(cid:3) 60(cid:3)
Function Values of , ,and
In Examples 1 and 3, we found the trigonometric function values of an
acute angle of a right triangle when the lengths of the three sides were
known.In most situations,we are asked to find the function values when
the measure of the acute angle is given.For certain special angles such as
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Chapter 5 • The Trigonometric Functions
30(cid:1), 45(cid:1), and 60(cid:1), which are frequently seen in applications, we can use
geometry to determine the function values.
A right triangle with a 45(cid:1) angle actually has two 45(cid:1) angles.Thus the
triangle is isosceles, and the legs are the same length. Let’s consider such
a triangle whose legs have length 1. Then we can find the length of its
hypotenuse,c,using the Pythagorean theorem as follows:
12 (cid:8) 12 (cid:1) c2, or c2 (cid:1) 2, or c (cid:1) (cid:3)2.
Such a triangle is shown below. From this diagram, we can easily deter-
mine the trigonometric function values of 45(cid:1).
opp 1 (cid:3)2
sin 45(cid:1) (cid:1) (cid:1) (cid:1) (cid:4) 0.7071,
45(cid:1) hyp (cid:3)2 2
adj 1 (cid:3)2
(cid:3)2 cos 45(cid:1) (cid:1) (cid:1) (cid:1) (cid:4) 0.7071,
1 hyp (cid:3)2 2
opp 1
tan 45(cid:1) (cid:1) (cid:1) (cid:1) 1
45(cid:1) adj 1
1
It is sufficient to find only the function values of the sine, cosine, and
tangent,since the others are their reciprocals.
It is also possible to determine the function values of 30(cid:1) and 60(cid:1).A
right triangle with 30(cid:1) and 60(cid:1) acute angles is half of an equilateral tri-
angle,as shown in the following figure.Thus if we choose an equilateral
triangle whose sides have length 2 and take half of it, we obtain a right
triangle that has a hypotenuse of length 2 and a leg of length 1.The other
leg has length a,which can be found as follows:
a2 (cid:8) 12 (cid:1) 22
a2 (cid:8) 1 (cid:1) 4
2 a 30(cid:1) 2 a2 (cid:1) 3
a (cid:1) (cid:3)3.
60(cid:1)
1 1
We can now determine the function values of30(cid:1)and 60(cid:1):
1 (cid:3)3
sin 30(cid:1) (cid:1) (cid:1) 0.5, sin 60(cid:1) (cid:1) (cid:4) 0.8660,
2 2
(cid:3)3 30(cid:1) 2 cos 30(cid:1) (cid:1) (cid:3)3 (cid:4) 0.8660, cos 60(cid:1) (cid:1) 1 (cid:1) 0.5,
2 2
60(cid:1) 1 (cid:3)3 (cid:3)3
1 tan 30(cid:1) (cid:1) (cid:1) (cid:4) 0.5774, tan 60(cid:1) (cid:1) (cid:1) (cid:3)3 (cid:4) 1.7321.
(cid:3)3 3 1
Since we will often use the function values of 30(cid:1),45(cid:1),and 60(cid:1),either the
triangles that yield them or the values themselves should be memorized.
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Section 5.1 • Trigonometric Functions of Acute Angles
30(cid:3) 45(cid:3) 60(cid:3)
sin 1(cid:7)2 (cid:3)2(cid:7)2 (cid:3)3(cid:7)2
cos (cid:3)3(cid:7)2 (cid:3)2(cid:7)2 1(cid:7)2
tan (cid:3)3(cid:7)3 1 (cid:3)3
Let’s now use what we have learned about trigonometric functions of
special angles to solve problems. We will consider such applications in
greater detail in Section 5.2.
EXAMPLE 4 Height ofa Hot-air Balloon. As a hot-air balloon began to
rise, the ground crew drove 1.2 mi to an observation station. The initial
observation from the station estimated the angle between the ground and
the line ofsight to the balloon to be 30(cid:1).Approximately how high was the
balloon at that point?(We are assuming that the wind velocity was low
and that the balloon rose vertically for the first few minutes.)
Solution We begin with a drawing ofthe situation.We know the measure
ofan acute angle and the length ofits adjacent side.
hh
30°
1.2 mi
Since we want to determine the length of the opposite side,we can use
the tangent ratio,or the cotangent ratio.Here we use the tangent ratio:
opp h
tan 30(cid:1) (cid:1) (cid:1)
adj 1.2
1.2 (cid:5)tan 30(cid:6)(cid:1) (cid:1) h
1.2 (cid:3)3 (cid:1) h Substituting;tan 30(cid:3)(cid:4)(cid:3)3
3 3
0.7 (cid:4) h.
The balloon is approximately 0.7 mi,or 3696 ft,high.
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Chapter 5 • The Trigonometric Functions
Function Values of Any Acute Angle
Historically, the measure of an angle has been expressed in degrees,
minutes, and seconds. One minute, denoted 1(cid:9), is such that 60(cid:9) (cid:1) 1(cid:1),
or 1(cid:9) (cid:1) 1 (cid:11) (cid:1)1(cid:1)(cid:2). One second, denoted 1(cid:10), is such that 60(cid:10) (cid:1) 1(cid:9), or
60
1(cid:10) (cid:1) 1 (cid:11) (cid:1)1(cid:9)(cid:2). Then 61 degrees, 27 minutes, 4 seconds could be written as
60
61(cid:1)27(cid:9)4(cid:10). This D(cid:3)M(cid:5)S(cid:6) form was common before the widespread use of
scientific calculators. Now the preferred notation is to express fractional
parts of degrees in decimal degree form. Although the D(cid:1)M(cid:9)S(cid:10) notation
is still widely used in navigation, we will most often use the decimal form
in this text.
Most calculators can convert D(cid:1)M(cid:9)S(cid:10) notation to decimal degree
notation and vice versa.Procedures among calculators vary.
GCM EXAMPLE 5 Convert 5(cid:1)42(cid:9)30(cid:10)to decimal degree notation.
Normal Sci Eng
Float 0123456789
Radian Degree Solution We can use a graphing calculator set in DEGREEmode to convert
Func Par Pol Seq between D(cid:1)M(cid:9)S(cid:10)form and decimal degree form.(See window at left.)
Connected Dot
Sequential Simul To convert D(cid:1)M(cid:9)S(cid:10) form to decimal degree form, we enter 5(cid:1)42(cid:9)30(cid:10)
Real a(cid:8)bi reˆθi using the ANGLEmenu for the degree and minute symbols and ALPHA (cid:8)
Full Horiz G–T
for the symbol representing seconds.Pressing ENTER gives us
5(cid:1)42(cid:9)30(cid:10) (cid:4) 5.71(cid:1),
rounded to the nearest hundredth ofa degree.
5°42'30"
5.708333333
Without a calculator,we can convert as follows:
5(cid:1)42(cid:9)30(cid:10) (cid:1) 5(cid:1) (cid:8) 42(cid:9) (cid:8) 30(cid:10)
(cid:1) 5(cid:1) (cid:8) 42(cid:9) (cid:8) 30(cid:9) 1(cid:6)(cid:4) 1(cid:5);30(cid:6)(cid:4)30(cid:5)
60 60 60
(cid:1) 5(cid:1) (cid:8) 42.5(cid:9) 30(cid:5)(cid:4)0.5(cid:5)
60
(cid:1) 5(cid:1) (cid:8) 42.5(cid:1) 1(cid:5)(cid:4) 1(cid:3);42.5(cid:5)(cid:4)42.5(cid:3)
60 60 60
42.5(cid:3)
(cid:4) 5.71(cid:1). (cid:4)0.71(cid:3)
60
GCM EXAMPLE 6 Convert 72.18(cid:1)to D(cid:1)M(cid:9)S(cid:10)notation.
Solution To convert decimal degree form to D(cid:1)M(cid:9)S(cid:10)form,we enter 72.18
and access the (cid:1)DMSfeature in the ANGLEmenu.The result is
72.18(cid:1) (cid:1) 72(cid:1)10(cid:9)48(cid:10).
72.18 DMS
72°10'48"
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Section 5.1 • Trigonometric Functions of Acute Angles
Without a calculator,we can convert as follows:
72.18(cid:1) (cid:1) 72(cid:1) (cid:8) 0.18 (cid:12) 1(cid:1)
(cid:1) 72(cid:1) (cid:8) 0.18 (cid:12) 60(cid:9) 1(cid:3)(cid:4)60(cid:5)
(cid:1) 72(cid:1) (cid:8) 10.8(cid:9)
(cid:1) 72(cid:1) (cid:8) 10(cid:9) (cid:8) 0.8 (cid:12) 1(cid:9)
(cid:1) 72(cid:1) (cid:8) 10(cid:9) (cid:8) 0.8 (cid:12) 60(cid:10) 1(cid:5)(cid:4)60(cid:6)
(cid:1) 72(cid:1) (cid:8) 10(cid:9) (cid:8) 48(cid:10)
(cid:1) 72(cid:1)10(cid:9)48(cid:10).
So far we have measured angles using degrees. Another useful unit
for angle measure is the radian,which we will study in Section 5.4.Calcu-
lators work with either degrees or radians.Be sure to use whichever mode is
appropriate.In this section,we use the degree mode.
Keep in mind the difference between an exact answer and an approxi-
mation.For example,
(cid:3)3
sin 60(cid:1) (cid:1) . This is exact!
2
But using a calculator,you get an answer like
sin 60(cid:1) (cid:4) 0.8660254038. This is an approximation!
Calculators generally provide values only of the sine, cosine, and
tangent functions. You can find values of the cosecant, secant, and co-
tangent by taking reciprocals of the sine, cosine, and tangent functions,
respectively.
GCM EXAMPLE 7 Find the trigonometric function value, rounded to four
decimal places,of each of the following.
a) tan 29.7(cid:1) b) sec 48(cid:1) c) sin 84(cid:1)10(cid:9)39(cid:10)
Solution
a) We check to be sure that the calculator is in DEGREEmode.The function
value is
tan 29.7(cid:1) (cid:4) 0.5703899297
(cid:4) 0.5704. Rounded to four decimal places
b) The secant function value can be found by taking the reciprocal of the
cosine function value:
1
sec 48(cid:1) (cid:1) (cid:4) 1.49447655 (cid:4) 1.4945.
cos 48(cid:1)
c) We enter sin 84(cid:1)10(cid:9)39(cid:10).The result is
sin 84(cid:1)10(cid:9)39(cid:10) (cid:4) 0.9948409474 (cid:4) 0.9948.
We can use the TABLE feature on a graphing calculator to find an angle
for which we know a trigonometric function value.