Section 4.4 Trigonometric Functions of Any Angle 537
Section 4.4 Trigonometric Functions of Any Angle
SECTION 4.4
Trigonometric Functions of Any Angle
Cycles govern many aspects of
lifeheartbeats, sleep patterns,
seasons, and tides all follow
regular,
predictable
cycles.
Because of their periodic nature,
trigonometric
functions
are
used to model phenomena that
occur in cycles. It is helpful to
apply these models regardless of
whether we think of the domains
of trigonometric functions as sets
of real numbers or sets of angles.
In order to understand and use
models for cyclic phenomena from
an angle perspective, we need to
move beyond right triangles.
Objectives
??? Use the de?nitions of
???
???
???
537
trigonometric functions
of any angle.
Use the signs of the
trigonometric functions.
Find reference angles.
Use reference angles to
evaluate trigonometric
functions.
Trigonometric Functions
of Any Angle
???
In the last section, we evaluated
trigonometric functions of acute
angles, such as that shown in Figure 4.41(a). Note that this angle is in standard
position. The point P = (x, y) is a point r units from the origin on the terminal
side of u. A right triangle is formed by drawing a line segment from P = (x, y)
perpendicular to the x@axis. Note that y is the length of the side opposite u and x is
the length of the side adjacent to u.
Use the de?nitions of
trigonometric functions
of any angle.
y
y
y
u
P (x, y)
P (x, y)
y
(a) u lies in
quadrant I.
r
u
u
x
y
r
y
x
x
(b) u lies in
quadrant II.
u
x
x
r
y
x
x
r
y
P (x, y)
x
P (x, y)
(c) u lies in
quadrant III.
(d) u lies in
quadrant IV.
FIGURE 4.41
Figures 4.41(b), (c), and (d) show angles in standard position, but they are not
acute. We can extend our de?nitions of the six trigonometric functions to include
such angles, as well as quadrantal angles. (Recall that a quadrantal angle has its
terminal side on the x@axis or y@axis; such angles are not shown in Figure 4.41.) The
point P = (x, y) may be any point on the terminal side of the angle u other than the
origin, (0, 0).
538
Chapter 4 Trigonometric Functions
y
De?nitions of Trigonometric Functions of Any Angle
P (x, y)
r
y
u
x
x
FIGURE 4.41(a) u lies in quadrant I.
(repeated)
Let u be any angle in standard position and let P = (x, y) be a point on the
terminal side of u. If r = 2x2 + y 2 is the distance from (0, 0) to (x, y), as shown
in Figure 4.41 on the previous page, the six trigonometric functions of U are
de?ned by the following ratios:
y
r
sin u=
csc u= , y 0
r
y
x
r
cos u=
sec u= , x 0
r
x
x
y
tan u= , x 0
cot u= , y 0.
y
x
GREAT QUESTION!
The ratios in the second column
are the reciprocals of the
corresponding ratios in the
first column.
Is there a way to make it
easier for me to remember the
de?nitions of trigonometric
functions of any angle?
Yes. If u is acute, we have the right
triangle shown in Figure 4.41(a).
In this situation, the de?nitions
in the box are the right triangle
de?nitions of the trigonometric
functions. This should make it
easier for you to remember the
six de?nitions.
Because the point P = (x, y) is any point on the terminal side of u other than
the origin, (0, 0), r = 2x2 + y 2 cannot be zero. Examine the six trigonometric
functions de?ned above. Note that the denominator of the sine and cosine functions
is r. Because r ? 0, the sine and cosine functions are de?ned for any angle u. This
is not true for the other four trigonometric functions. Note that the denominator of
y
r
the tangent and secant functions is x: tan u = and sec u = . These functions are
x
x
not de?ned if x = 0. If the point P = (x, y) is on the y@axis, then x = 0. Thus, the
tangent and secant functions are unde?ned for all quadrantal angles with terminal
sides on the positive or negative y@axis. Likewise, if P = (x, y) is on the x@axis, then
x
y = 0, and the cotangent and cosecant functions are unde?ned: cot u = and
y
r
csc u = . The cotangent and cosecant functions are unde?ned for all quadrantal
y
angles with terminal sides on the positive or negative x@axis.
EXAMPLE 1
Evaluating Trigonometric Functions
Let P = (-3, -5) be a point on the terminal side of u. Find each of the six
trigonometric functions of u.
y
5
SOLUTION
u
?5
5
x
The situation is shown in Figure 4.42. We need values for x, y, and r to evaluate all six
trigonometric functions. We are given the values of x and y. Because P = (-3, -5)
is a point on the terminal side of u, x = -3 and y = -5. Furthermore,
r = 2x2 + y 2 = 2(-3)2 + (-5)2 = 29 + 25 = 234.
r
Now that we know x, y, and r, we can ?nd the six trigonometric functions of u.
Where appropriate, we will rationalize denominators.
?5
P = (?3, ? 5)
x = ?3
FIGURE 4.42
y = ?5
sin u =
y
-5
5 # 234
5234
=
= = r
34
234
234 234
-3
x
3 # 234
3234
=
= = r
34
234
234 234
y
-5
5
tan u = =
=
x
-3
3
cos u =
csc u =
234
r
234
=
= y
-5
5
234
r
234
=
= x
-3
3
-3
x
3
cot u = =
=
y
-5
5
sec u =
Check Point 1 Let P = (1, -3) be a point on the terminal side of u. Find each
of the six trigonometric functions of u.
Section 4.4 Trigonometric Functions of Any Angle
539
How do we ?nd the values of the trigonometric functions for a quadrantal angle?
First, draw the angle in standard position. Second, choose a point P on the angles
terminal side. The trigonometric function values of u depend only on the size of u and
not on the distance of point P from the origin. Thus, we will choose a point that is 1 unit
from the origin. Finally, apply the de?nitions of the appropriate trigonometric functions.
EXAMPLE 2
Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the sine function and the tangent function at the following
four quadrantal angles:
p
3p
a. u = 0? = 0 b. u = 90? =
c. u = 180? = p d. u = 270? =
.
2
2
SOLUTION
x=1
y
u = 0?
y=0
P = (1, 0)
?1
x
1
r=1
FIGURE 4.43
y
x=0
y=1
P = (0, 1)
1
r=1
u = 90?
?1
x
1
FIGURE 4.44
y
x = ?1
y=0
1
u = 180?
P = (?1, 0)
?1
1
x
r=1
FIGURE 4.45
y
u = 270?
?1
r=1
x
1
?1
P = (0, ?1)
x=0
FIGURE 4.46
y = ?1
a. If u = 0? = 0 radians, then the terminal side of the angle is on the positive
x@axis. Let us select the point P = (1, 0) with x = 1 and y = 0. This point
is 1 unit from the origin, so r = 1. Figure 4.43 shows values of x, y, and r
corresponding to u = 0? or 0 radians. Now that we know x, y, and r, we can
apply the de?nitions of the sine and tangent functions.
y
0
sin 0? = sin 0 = = = 0
r
1
y
0
tan 0? = tan 0 = = = 0
x
1
p
b. If u = 90? = radians, then the terminal side of the angle is on the positive
2
y@axis. Let us select the point P = (0, 1) with x = 0 and y = 1. This point
is 1 unit from the origin, so r = 1. Figure 4.44 shows values of x, y, and r
p
corresponding to u = 90? or . Now that we know x, y, and r, we can apply
2
the de?nitions of the sine and tangent functions.
y
1
p
sin 90? = sin = = = 1
r
2
1
y
1
p
tan 90? = tan = =
x
2
0
Because division by 0 is unde?ned, tan 90 is unde?ned.
c. If u = 180? = p radians, then the terminal side of the angle is on the
negative x@axis. Let us select the point P = (-1, 0) with x = -1 and y = 0.
This point is 1 unit from the origin, so r = 1. Figure 4.45 shows values of
x, y, and r corresponding to u = 180? or p. Now that we know x, y, and r,
we can apply the de?nitions of the sine and tangent functions.
y
0
sin 180? = sin p = = = 0
r
1
y
0
tan 180? = tan p = =
= 0
x
-1
3p
d. If u = 270? =
radians, then the terminal side of the angle is on the negative
2
y@axis. Let us select the point P = (0, -1) with x = 0 and y = -1. This
point is 1 unit from the origin, so r = 1. Figure 4.46 shows values of x, y,
3p
and r corresponding to u = 270? or
. Now that we know x, y, and r, we
2
can apply the de?nitions of the sine and tangent functions.
DISCOVERY
y
-1
3p
= =
= -1
Try ?nding tan 90 and tan 270
r
2
1
with your calculator. Describe
y
-1
3p
tan 270? = tan
= =
what occurs.
x
2
0
Because division by 0 is unde?ned, tan 270 is unde?ned.
sin 270? = sin
540
Chapter 4 Trigonometric Functions
Check Point 2 Evaluate, if possible, the cosine function and the cosecant
function at the following four quadrantal angles:
p
3p
a. u = 0? = 0 b. u = 90? =
c. u = 180? = p d. u = 270? =
.
2
2
???
The Signs of the Trigonometric Functions
Use the signs of the
trigonometric functions.
y
Quadrant II
sine and
cosecant
positive
Quadrant I
All
functions
positive
x
Quadrant III
tangent and
cotangent
positive
Quadrant IV
cosine and
secant
positive
FIGURE 4.47 The signs of the
trigonometric functions
In Example 2, we evaluated trigonometric functions of quadrantal angles. However,
we will now return to the trigonometric functions of nonquadrantal angles. If U is not
a quadrantal angle, the sign of a trigonometric function depends on the quadrant in
which U lies. In all four quadrants, r is positive. However, x and y can be positive or
negative. For example, if u lies in quadrant II, x is negative and y is positive. Thus, the
y
r
only positive ratios in this quadrant are and its reciprocal, . These ratios are the
y
r
function values for the sine and cosecant, respectively. In short, if u lies in quadrant II,
sin u and csc u are positive. The other four trigonometric functions are negative.
Figure 4.47 summarizes the signs of the trigonometric functions. If u lies in
quadrant I, all six functions are positive. If u lies in quadrant II, only sin u and csc u
are positive. If u lies in quadrant III, only tan u and cot u are positive. Finally, if u lies
in quadrant IV, only cos u and sec u are positive. Observe that the positive functions
in each quadrant occur in reciprocal pairs.
GREAT QUESTION!
Is there a way to remember the signs of the trigonometric functions?
Heres a phrase that may be helpful:
A
Trig
Smart
All trig functions
are positive in
QI.
EXAMPLE 3
Sine and its
reciprocal, cosecant,
are positive in QII.
Tangent and its
reciprocal, cotangent,
are positive in QIII.
Class.
Cosine and its
reciprocal, secant,
are positive in QIV.
Finding the Quadrant in Which an Angle Lies
If tan u 6 0 and cos u 7 0, name the quadrant in which angle u lies.
SOLUTION
When tan u 6 0, u lies in quadrant II or IV. When cos u 7 0, u lies in quadrant
I or IV. When both conditions are met (tan u 6 0 and cos u 7 0), u must lie in
quadrant IV.
Check Point 3 If sin u 6 0 and cos u 6 0, name the quadrant in which angle
u lies.
EXAMPLE 4
Evaluating Trigonometric Functions
2
3
Given tan u = - and cos u 7 0, ?nd cos u and csc u.
SOLUTION
Because the tangent is negative and the cosine is positive, u lies in quadrant IV.
This will help us to determine whether the negative sign in tan u = - 23 should
be associated with the numerator or the denominator. Keep in mind that in
quadrant IV, x is positive and y is negative. Thus,
In quadrant IV, y is negative.
y
C2
2
tan u=C = = .
x
3
3
Section 4.4 Trigonometric Functions of Any Angle
(See Figure 4.48.) Thus, x = 3 and y = -2. Furthermore,
y
5
u
r = 2x2 + y 2 = 232 + (-2)2 = 29 + 4 = 213.
Now that we know x, y, and r, we can ?nd cos u and csc u.
r = 13
?5
5
?5
541
x
P = (3, ? 2)
x=3
cos u =
3
x
3 # 213
3213
=
=
=
r
13
213
213 213
csc u =
213
r
213
=
= y
-2
2
Check Point 4 Given tan u = - 13 and cos u 6 0, ?nd sin u and sec u.
y = ?2
2
3
FIGURE 4.48 tan u = - and cos u 7 0
In Example 4, we used the quadrant in which u lies to determine whether a
negative sign should be associated with the numerator or the denominator. Heres a
situation, similar to Example 4, where negative signs should be associated with both
the numerator and the denominator:
tan u =
3
5
and cos u 6 0.
Because the tangent is positive and the cosine is negative, u lies in quadrant III. In
quadrant III, x is negative and y is negative. Thus,
y
3
C3
tan u= = =
.
x
5
C5
???
Find reference angles.
We see that x = ?5
and y = ?3.
Reference Angles
We will often evaluate trigonometric functions of positive angles greater than 90
and all negative angles by making use of a positive acute angle. This positive acute
angle is called a reference angle.
De?nition of a Reference Angle
Let u be a nonacute angle in standard position that lies in a quadrant. Its reference
angle is the positive acute angle u? formed by the terminal side of u and the x@axis.
Figure 4.49 shows the reference angle for u lying in quadrants II, III, and IV.
Notice that the formula used to ?nd u?, the reference angle, varies according to the
quadrant in which u lies. You may ?nd it easier to ?nd the reference angle for a given
angle by making a ?gure that shows the angle in standard position. The acute angle
formed by the terminal side of this angle and the x@axis is the reference angle.
y
u?
y
u
u
x
x
u?
FIGURE 4.49 Reference angles, u?, for
positive angles, u, in quadrants II, III,
and IV
If 90? ? u ? 180?,
then u ? ? 180? ? u.
y
If 180? ? u ? 270?,
then u ? ? u ? 180?.
u
u?
If 270? ? u ? 360?,
then u ? ? 360? ? u.
x
................
................
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