Download Section 4.4

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Section 4.4 Trigonometric Functions of Any Angle
73. Standing under this arch, I can determine its height by
measuring the angle of elevation to the top of the arch and
my distance to a point directly under the arch.
501
Preview Exercises
Exercises 81–83 will help you prepare for the material covered in
the next section. Use these figures to solve Exercises 81–82.
y
y
P (x, y)
P (x, y)
r
y
r
y
u
u
x
x
x
x
(b) u lies in
quadrant II.
(a) u lies in
quadrant I.
81. a. Write a ratio that expresses sin u for the right triangle in
Figure (a).
b. Determine the ratio that you wrote in part (a) for Figure (b)
with x = - 3 and y = 4. Is this ratio positive or negative?
Delicate Arch in Arches National Park, Utah
In Exercises 74–77, determine whether each statement is true or
false. If the statement is false, make the necessary change(s) to
produce a true statement.
74.
76.
78.
79.
80.
tan 45°
75. tan2 15° - sec2 15° = - 1
= tan 3°
tan 15°
77. tan2 5° = tan 25°
sin 45° + cos 45° = 1
Explain why the sine or cosine of an acute angle cannot be
greater than or equal to 1.
Describe what happens to the tangent of an acute angle as
the angle gets close to 90°. What happens at 90°?
From the top of a 250-foot lighthouse, a plane is sighted
overhead and a ship is observed directly below the plane.
The angle of elevation of the plane is 22° and the angle of
depression of the ship is 35°. Find a. the distance of the ship
from the lighthouse; b. the plane’s height above the water.
Round to the nearest foot.
Section
4.4
Objectives
� Use the definitions of
�
�
�
trigonometric functions of
any angle.
Use the signs of the
trigonometric functions.
Find reference angles.
Use reference angles to
evaluate trigonometric
functions.
82. a. Write a ratio that expresses cos u for the right triangle in
Figure (a).
b. Determine the ratio that you wrote in part (a) for Figure (b)
with x = - 3 and y = 5. Is this ratio positive or negative?
83. Find the positive angle u¿ formed by the terminal side of u
and the x-axis.
a.
y
u 345
x
u
b.
y
5p
u
u 6
x
Trigonometric Functions of Any Angle
C
ycles
govern
many
aspects of life—heartbeats, 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
trigonomentric 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.
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�
Trigonometric Functions of Any Angle
Use the definitions
of 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 = 1x, y2 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 = 1x, y2 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.
y
y
y
P (x, y)
u
x
x
P (x, y)
r
y
y
r
y
x
(a) u lies in
quadrant I.
Figure 4.41
Study Tip
If u is acute, we have the right triangle
shown in Figure 4.41(a). In this
situation, the definitions in the box
are the right triangle definitions of
the trigonometric functions. This
should make it easier for you to
remember the six definitions.
x
x
r
r
u
u
x
y
u
y
P (x, y)
x
x
P (x, y)
(b) u lies in
quadrant II.
(c) u lies in
quadrant III.
(d) u lies in
quadrant IV.
Figures 4.41(b), (c), and (d) show angles in standard position, but they are not
acute. We can extend our definitions 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 = 1x, y2
may be any point on the terminal side of the angle u other than the origin, (0, 0).
Definitions of Trigonometric Functions of Any Angle
Let u be any angle in standard position and let P = 1x, y2 be a point on the
terminal side of u. If r = 3x2 + y2 is the distance from (0, 0) to 1x, y2, as shown in
Figure 4.41, the six trigonometric functions of U are defined by the following ratios:
y
r
x
cos u=
r
y
tan u= , x 0
x
sin u=
r
,y0
y
r
sec u= , x 0
x
x
cot u= , y 0.
y
csc u=
The ratios in the second column
are the reciprocals of the
corresponding ratios in the
first column.
Because the point P = 1x, y2 is any point on the terminal side of u other than
the origin, (0, 0), r = 3x2 + y2 cannot be zero. Examine the six trigonometric
functions defined above. Note that the denominator of the sine and cosine
functions is r. Because r Z 0, the sine and cosine functions are defined for any
angle u. This is not true for the other four trigonometric functions. Note that the
y
r
denominator of the tangent and secant functions is x: tan u =
and sec u = .
x
x
These functions are not defined if x = 0. If the point P = 1x, y2 is on the y-axis,
then x = 0. Thus, the tangent and secant functions are undefined for all quadrantal
angles with terminal sides on the positive or negative y-axis. Likewise, if P = 1x, y2
is on the x-axis, then y = 0, and the cotangent and cosecant functions are
x
r
undefined: cot u =
and csc u = . The cotangent and cosecant functions
y
y
are undefined for all quadrantal angles with terminal sides on the positive or
negative x-axis.
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Section 4.4 Trigonometric Functions of Any Angle
503
Evaluating Trigonometric Functions
EXAMPLE 1
Let P = 1- 3, -52 be a point on the terminal side of u. Find each of the six trigonometric functions of u.
y
Solution 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 = 1- 3, - 52 is a point on the terminal side of u, x = - 3 and y = - 5.
Furthermore,
5
u
−5
5
r
Now that we know x, y, and r, we can find the six trigonometric functions of u.
Where appropriate, we will rationalize denominators.
−5
P = (−3, − 5)
x = −3
r = 3x2 + y2 = 31-322 + 1- 522 = 29 + 25 = 234.
x
sin u =
y
-5
5 # 234
5234
=
= = r
34
234
234 234
csc u =
r
234
234
=
= y
-5
5
cos u =
x
-3
3 # 234
3234
=
= = r
34
234
234 234
sec u =
r
234
234
=
= x
-3
3
tan u =
y
-5
5
=
=
x
-3
3
cot u =
x
-3
3
=
=
y
-5
5
y = −5
Figure 4.42
Let P = 11, - 32 be a point on the terminal side of u. Find each
of the six trigonometric functions of u.
Check Point
1
How do we find the values of the trigonometric functions for a quadrantal
angle? First, draw the angle in standard position. Second, choose a point P on the
angle’s 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 definitions 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
y
u = 0
−1
x=1
y=0
P = (1, 0)
1
x
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 = 11, 02 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 definitions of the sine and tangent functions.
r=1
Figure 4.43
sin 0° = sin 0 =
y
0
= = 0
r
1
tan 0° = tan 0 =
y
0
= = 0
x
1
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p
radians, then the terminal side of the angle is on the positive
2
y-axis. Let us select the point P = 10, 12 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 the
2
definitions of the sine and tangent functions.
y
b. If u = 90° =
x=0
y=1
P = (0, 1)
1
r=1
u = 90⬚
−1
x
1
y
p
1
=
= = 1
r
2
1
y
1
p
=
tan 90° = tan =
x
2
0
sin 90° = sin
Figure 4.44
Because division by 0 is undefined, tan 90° is undefined.
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- 1, 02 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
definitions of the sine and tangent functions.
y
x = −1
y=0
1
u = 180⬚
P = (−1, 0)
−1
1
x
sin 180° = sin p =
y
0
= = 0
r
1
tan 180° = tan p =
y
0
=
= 0
x
-1
r=1
Figure 4.45
u = 270⬚
−1
r=1
x
1
−1
3p
radians, then the terminal side of the angle is on the negative
2
y-axis. Let us select the point P = 10, -12 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, and r
3p
corresponding to u = 270° or
. Now that we know x, y, and r, we can apply
2
the definitions of the sine and tangent functions.
d. If u = 270° =
y
P = (0, −1)
x=0
Discovery
y
3p
-1
=
=
= -1
Try finding tan 90° and
r
2
1
tan 270° with your calcuy
-1
3p
tan 270° = tan
=
=
lator. Describe what occurs.
x
2
0
sin 270° = sin
y = −1
Figure 4.46
Because division by 0 is undefined, tan 270° is undefined.
2
�
Check Point 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
Use the signs of the
trigonometric functions.
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
r
y
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.
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Section 4.4 Trigonometric Functions of Any Angle
505
Study Tip
Here’s a phrase to help you remember the signs of the trig functions:
A
Smart
All trig functions
are positive in
QI.
Sine and its
reciprocal, cosecant,
are positive in QII.
Trig
Class.
Tangent and its
reciprocal, cotangent,
are positive in QIII.
Cosine and its
reciprocal, secant,
are positive in QIV.
Finding the Quadrant in Which an Angle Lies
EXAMPLE 3
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.
Evaluating Trigonometric Functions
EXAMPLE 4
Given tan u = - 23 and cos u 7 0, find 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,
y
5
r = 兹13
u
−5
5
−5
P = (3, − 2)
x=3
y
–2
2
tan u=– = = .
x
3
3
(See Figure 4.48.) Thus, x = 3 and y = - 2. Furthermore,
r = 3x2 + y2 = 332 + 1- 222 = 29 + 4 = 213.
y = −2
Figure 4.48 tan u = - 23 and
cos u 7 0
In quadrant IV, y is negative.
x
Now that we know x, y, and r, we can find cos u and csc u.
cos u =
x
3
3 # 213
3213
=
=
=
r
13
213
213 213
Check Point
4
csc u =
r
213
213
=
= y
-2
2
Given tan u = - 13 and cos u 6 0, find sin u and sec u.
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. Here’s 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
–3
tan u= = =
.
x
5
–5
We see that x = −5
and y = −3.
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�
Find reference angles.
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.
Definition 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 find u¿, the reference angle, varies according to the
quadrant in which u lies. You may find it easier to find the reference angle for a
given angle by making a figure 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
y
u
y
u
u
x
u
x
x
u
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.
EXAMPLE 5
If 180 u 270,
then u u 180.
If 270 u 360,
then u 360 u.
Finding Reference Angles
Find the reference angle, u¿, for each of the following angles:
a. u = 345°
Solution
b. u =
5p
6
c. u = - 135°
d. u = 2.5.
y
a. A 345° angle in standard position is shown in
Figure 4.50. Because 345° lies in quadrant
IV, the reference angle is
u¿ = 360° - 345° = 15°.
p
5p
3p
=
lies between
and
6
2
6
6p
5p
p =
,u =
lies in quadrant II. The
6
6
angle is shown in Figure 4.51. The reference
angle is
u 345
x
u 15
Figure 4.50
b. Because
u¿ = p -
Discovery
Solve part (c) by first finding a positive
coterminal angle for - 135° less than
360°. Use the positive coterminal
angle to find the reference angle.
6p
5p
p
5p
=
= .
6
6
6
6
c. A -135° angle in standard position is
shown in Figure 4.52. The figure indicates
that the positive acute angle formed by the
terminal side of u and the x-axis is 45°. The
reference angle is
u¿ = 45°.
y
5p
u 6
p
u 6
x
Figure 4.51
y
u 45
Figure 4.52
x
u 135
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Section 4.4 Trigonometric Functions of Any Angle
p
L 1.57 and p L 3.14. This means that
2
u = 2.5 is in quadrant II, shown in Figure 4.53. The reference angle is
y
u 艐 0.64
507
d. The angle u = 2.5 lies between
u 2.5
x
u¿ = p - 2.5 L 0.64.
Figure 4.53
5
Check Point
a. u = 210°
Find the reference angle, u¿, for each of the following angles:
b. u =
7p
4
c. u = - 240°
d. u = 3.6.
Finding reference angles for angles that are greater than 360° 12p2 or less than
- 360° 1- 2p2 involves using coterminal angles. We have seen that coterminal angles
have the same initial and terminal sides. Recall that coterminal angles can be obtained
by increasing or decreasing an angle’s measure by an integer multiple of 360° or 2p.
Finding Reference Angles for Angles Greater Than 360° (2P)
or Less Than 360° (2P)
1. Find a positive angle a less than 360° or 2p that is coterminal with the given
angle.
2. Draw a in standard position.
3. Use the drawing to find the reference angle for the given angle. The positive
acute angle formed by the terminal side of a and the x-axis is the reference
angle.
EXAMPLE 6
Finding Reference Angles
Find the reference angle for each of the following angles:
8p
13p
a. u = 580°
b. u =
c. u = .
3
6
Solution
a. For a 580° angle, subtract 360° to find a positive coterminal angle less than 360°.
y
580° - 360° = 220°
Figure 4.54 shows a = 220° in standard position.
Because 220° lies in quadrant III, the reference
angle is
a¿ = 220° - 180° = 40°.
a 220
a 40
x
Figure 4.54
8p
2
b. For an
, or 2 p, angle, subtract 2p to find a positive coterminal angle less
3
3
than 2p.
8p
6p
2p
8p
- 2p =
=
3
3
3
3
y
2p
Figure 4.55 shows a =
in standard position.
3
2p
2p
Because
lies in quadrant II, the reference
p
a 3
a 3
3
x
angle is
2p
3p
2p
p
a¿ = p =
= .
Figure 4.55
3
3
3
3
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508 Chapter 4 Trigonometric Functions
Discovery
Solve part (c) using the coterminal
angle formed by adding 2p, rather
than 4p, to the given angle.
y
13p
1
, or -2 p, angle, add 4p to find a
6
6
positive coterminal angle less than 2p.
c. For a -
11p
a 6
p
a 6
13p
13p
24p
11p
+ 4p = +
=
6
6
6
6
x
Figure 4.56
11p
in standard position.
6
11p
Because
lies in quadrant IV, the reference angle is
6
Figure 4.56 shows a =
a¿ = 2p -
Check Point
a. u = 665°
�
Use reference angles to evaluate
trigonometric functions.
6
11p
12p
11p
p
=
= .
6
6
6
6
Find the reference angle for each of the following angles:
b. u =
15p
4
c. u = -
11p
.
3
Evaluating Trigonometric Functions Using Reference Angles
The way that reference angles are defined makes them useful in evaluating
trigonometric functions.
Using Reference Angles to Evaluate Trigonometric Functions
The values of the trigonometric functions of a given angle, u, are the same as the
values of the trigonometric functions of the reference angle, u¿, except possibly
for the sign. A function value of the acute reference angle, u¿, is always positive.
However, the same function value for u may be positive or negative.
For example, we can use a reference angle, u¿, to obtain an exact value for
tan 120°. The reference angle for u = 120° is u¿ = 180° - 120° = 60°. We know the
exact value of the tangent function of the reference angle: tan 60° = 23. We also
know that the value of a trigonometric function of a given angle, u, is the same as
that of its reference angle, u¿, except possibly for the sign. Thus, we can conclude that
tan 120° equals - 23 or 23.
What sign should we attach to 23? A 120° angle lies in quadrant II, where
only the sine and cosecant are positive. Thus, the tangent function is negative for a
120° angle. Therefore,
Prefix by a negative sign to
show tangent is negative in
quadrant II.
tan 120=–tan 60=–兹3.
The reference angle
for 120° is 60°.
In the previous section, we used two right triangles to find exact trigonometric
values of 30°, 45°, and 60°. Using a procedure similar to finding tan 120°, we can now
find the exact function values of all angles for which 30°, 45°, or 60° are reference angles.
A Procedure for Using Reference Angles to Evaluate
Trigonometric Functions
The value of a trigonometric function of any angle u is found as follows:
1. Find the associated reference angle, u¿, and the function value for u¿.
2. Use the quadrant in which u lies to prefix the appropriate sign to
the function value in step 1.
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Section 4.4 Trigonometric Functions of Any Angle
Discovery
EXAMPLE 7
Draw the two right triangles involving
30°, 45°, and 60°. Indicate the length
of each side. Use these lengths to
verify the function values for the
reference angles in the solution to
Example 7.
509
Using Reference Angles to Evaluate
Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric
functions:
4p
p
a. sin 135°
b. cos
c. cot a - b.
3
3
Solution
a. We use our two-step procedure to find sin 135°.
Step 1 Find the reference angle, U œ , and sin U œ . Figure 4.57 shows 135° lies in
quadrant II. The reference angle is
y
u¿ = 180° - 135° = 45°.
135
45
x
The function value for the reference angle is sin 45° =
22
.
2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the
function value in step 1. The angle u = 135° lies in quadrant II. Because
the sine is positive in quadrant II, we put a + sign before the function value of
the reference angle. Thus,
Figure 4.57 Reference angle
for 135°
The sine is positive
in quadrant II.
sin 135=±sin 45=
兹2
.
2
The reference angle
for 135° is 45°.
b. We use our two-step procedure to find cos
y
4p
3
x
p
3
4p
.
3
Step 1 Find the reference angle, U œ , and cos U œ . Figure 4.58 shows that
4p
u =
lies in quadrant III. The reference angle is
3
4p
3p
p
4p
- p =
= .
u¿ =
3
3
3
3
The function value for the reference angle is
Figure 4.58 Reference
cos
4p
angle for
3
p
1
= .
3
2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the
4p
function value in step 1. The angle u =
lies in quadrant III. Because only the
3
tangent and cotangent are positive in quadrant III, the cosine is negative in
this quadrant. We put a - sign before the function value of the reference
angle. Thus,
The cosine is negative
in quadrant III.
cos
1
p
4p
=–cos =– .
2
3
3
The reference angle
for 4p is p.
3
3
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510 Chapter 4 Trigonometric Functions
c. We use our two-step procedure to find cot a -
Step 1 Find the reference angle, U œ , and cot U œ . Figure 4.59 shows that
p
p
u = - lies in quadrant IV. The reference angle is u¿ = . The function value
3
3
p
23
for the reference angle is cot
=
.
3
3
y
x
p
3
p
3
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the
p
function value in step 1. The angle u = - lies in quadrant IV. Because only the
3
cosine and secant are positive in quadrant IV, the cotangent is negative in this
quadrant. We put a - sign before the function value of the reference angle. Thus,
Figure 4.59 Reference
angle for -
p
b.
3
p
3
The cotangent is
negative in quadrant IV.
p
p
兹3
cot a– b =–cot =–
.
3
3
3
The reference angle
for − p is p.
3 3
7
Check Point Use reference angles to find the exact value of the following
trigonometric functions:
a. sin 300°
b. tan
c. sec a -
5p
4
p
b.
6
In our final example, we use positive coterminal angles less than 2p to find the
reference angles.
EXAMPLE 8
Using Reference Angles to Evaluate
Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric
functions:
a. tan
14p
3
b. sec a -
17p
b.
4
Solution
14p
.
3
14p
Step 1 Find the reference angle, U œ , and tan U œ . Because the given angle,
3
2
or 4 p, exceeds 2p, subtract 4p to find a positive coterminal angle less than 2p.
3
14p
14p
12p
2p
u =
- 4p =
=
3
3
3
3
a. We use our two-step procedure to find tan
y
p
3
2p
3
x
Figure 4.60 Reference angle
for
2p
3
2p
Figure 4.60 shows u =
in standard position. The angle lies in quadrant II.
3
The reference angle is
2p
3p
2p
p
u¿ = p =
= .
3
3
3
3
p
= 23.
The function value for the reference angle is tan
3
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Section 4.4 Trigonometric Functions of Any Angle
511
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the
2p
function value in step 1. The coterminal angle u =
lies in quadrant II.
3
Because the tangent is negative in quadrant II, we put a - sign before the
function value of the reference angle. Thus,
The tangent is negative
in quadrant II.
tan
p
2p
14p
=tan
=–tan =–兹3 .
3
3
3
The reference angle
for 2p is p.
3
3
b. We use our two-step procedure to find sec a -
17p
b.
4
Step 1 Find the reference angle, U œ , and sec U œ . Because the given angle,
17p
1
or -4 p, is less than -2p, add 6p (three multiples of 2p) to find a
4
4
positive coterminal angle less than 2p.
u = -
Figure 4.61 shows u =
y
17p
17p
24p
7p
+ 6p = +
=
4
4
4
4
7p
in standard position. The angle lies in quadrant IV.
4
The reference angle is
7p
4
p
4
Figure 4.61 Reference angle
for
7p
4
x
u¿ = 2p -
8p
7p
p
7p
=
= .
4
4
4
4
The function value for the reference angle is sec
p
= 22.
4
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the
7p
function value in step 1. The coterminal angle u =
lies in quadrant IV.
4
Because the secant is positive in quadrant IV, we put a + sign before the
function value of the reference angle. Thus,
The secant is
positive in quadrant IV.
sec a–
17p
7p
p
b =sec
=± sec =兹2 .
4
4
4
The reference angle
for 7p is p.
4
4
Check Point
8
Use reference angles to find the exact value of each of the
following trigonometric functions:
a. cos
17p
6
b. sin a -
22p
b.
3
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512 Chapter 4 Trigonometric Functions
Study Tip
Evaluating trigonometric functions like those in Example 8 and Check Point 8 involves using a number of concepts, including finding
coterminal angles and reference angles, locating special angles, determining the signs of trigonometric functions in specific quadrants,
p
p
p
and finding the trigonometric functions of special angles a 30° = , 45° = , and 60° = b. To be successful in trigonometry, it is
6
4
3
often necessary to connect concepts. Here’s an early reference sheet showing some of the concepts you should have at your fingertips
(or memorized).
Degree and Radian Measures of Special and Quadrantal Angles
p
90 , 2
2p
120 , 3
3p
135 , 4
5p
150 , 6
7p
11p
7p
330 , 6
7p
315 , 4
4p
240 , 3
3p
270 , 2
3p
270 , 2
210 , 6
0 , 0
210 , 6
5p
225 , 4
5p
300 , 3
180 , p
45
30° U
1
30
P
6
5p
150 , 6
3p
135 , 4
2p
120 , 3
P
4
45° p
30 , 6
p
45 , 4
p
60
,
p
3
90 , 2
Signs of the Trigonometric
Functions
60° sin U
1
2
22
2
23
2
cos U
23
2
22
2
1
2
tan U
23
3
1
23
兹3
60
5p
300 , 3
7p
315 , 4
11p
330 , 6
0 , 0
Special Right Triangles and Trigonometric
Functions of Special Angles
1
2
p
30 , 6
180 , p
兹2
4p
240 , 3
5p
225 , 4
p
60 , 3
p
45 , 4
P
3
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
1
Using Reference Angles to
Evaluate Trigonometric Functions
Trigonometric Functions of Quadrantal Angles
U
sin U
cos U
tan U
0° 0
0
1
0
90° P
2
1
0
undefined
180° P
0
-1
0
270° 3P
2
-1
0
undefined
sin u=
cos u=
tan u=
sin u
cos u
tan u
+ or − in
determined by the
quadrant in which u lies and the sign
of the function in that quadrant.
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Section 4.4 Trigonometric Functions of Any Angle
513
Exercise Set 4.4
Practice Exercises
In Exercises 1–8, a point on the terminal side of angle u is given.
Find the exact value of each of the six trigonometric functions of u.
1. 1 -4, 32
2. 1- 12, 52
4. (3, 7)
7. 1- 2, -52
3. (2, 3)
5. 13, - 32
6. 15, -52
8. 1 - 1, -32
In Exercises 9–16, evaluate the trigonometric function at the
quadrantal angle, or state that the expression is undefined.
9. cos p
10. tan p
3p
13. tan
2
p
16. tan
2
12. csc p
p
15. cot
2
11. sec p
3p
14. cos
2
In Exercises 17–22, let u be an angle in standard position.
Name the quadrant in which u lies.
17. sin u 7 0, cos u 7 0
18. sin u 6 0, cos u 7 0
19. sin u 6 0, cos u 6 0
20. tan u 6 0, sin u 6 0
21. tan u 6 0, cos u 6 0
22. cot u 7 0, sec u 6 0
In Exercises 23–34, find the exact value of each of the remaining
trigonometric functions of u.
23. cos u = - 35 ,
u in quadrant III
24. sin u = - 12
13 ,
u in quadrant III
25. sin u =
26. cos u =
27. cos u =
28. cos u =
29. tan u =
31. tan u =
5
13 , u in quadrant II
4
5 , u in quadrant IV
8
17 , 270° 6 u 6 360°
1
3 , 270° 6 u 6 360°
- 23 , sin u 7 0
30.
4
32.
3 , cos u 6 0
33. sec u = - 3, tan u 7 0
tan u = - 13 ,
tan u =
5
12 ,
2p
3
7p
70. cot
4
7p
6
9p
72. tan
2
3p
4
9p
71. tan
4
67. sin
68. cos
69. csc
73. sin1 - 240°2
74. sin1 -225°2
75. tan a -
76. tan a -
77. sec 495°
78. sec 510°
p
b
6
19p
79. cot
6
35p
82. cos
6
17p
b
85. sin a 3
13p
3
17p
b
83. tan a 6
35p
b
86. sin a 6
80. cot
p
b
4
23p
4
11p
b
84. tan a 4
81. cos
Practice Plus
In Exercises 87–92, find the exact value of each expression. Write
the answer as a single fraction. Do not use a calculator.
p
p
3p
cos p - cos sin
3
3
2
p
p
88. sin cos 0 - sin cos p
4
6
11p
5p
11p
cos
+ cos
sin
89. sin
4
6
4
17p
5p
17p
cos
+ cos
sin
90. sin
3
4
3
87. sin
5p
6
5p
4
91. sin
3p
15p
5p
tan a b - cos a b
2
4
3
92. sin
3p
8p
5p
tan a b + cos a b
2
3
6
sin u 7 0
cos u 6 0
In Exercises 93–98, let
f1x2 = sin x, g1x2 = cos x, and h1x2 = 2x.
34. csc u = - 4, tan u 7 0
Find the exact value of each expression. Do not use a calculator.
In Exercises 35–60, find the reference angle for each angle.
35. 160°
38. 210°
7p
41.
4
5p
44.
7
47. - 335°
50. 5.5
17p
53.
6
17p
56.
3
25p
59. 6
36. 170°
39. 355°
5p
42.
4
37. 205°
40. 351°
5p
43.
6
45. - 150°
46. -250°
48. -359°
51. 565°
11p
54.
4
11p
57. 4
13p
60. 3
49. 4.7
52. 553°
23p
55.
4
17p
58. 6
93. f a
4p
p
4p
p
+ b + fa b + fa b
3
6
3
6
94. g a
5p
p
5p
p
+ b + ga b + ga b
6
6
6
6
95. 1h ⴰ g2a
96. 1h ⴰ f2a
11p
b
4
97. the average rate of change of f from x1 =
98. the average rate of change of g from x1 =
5p
3p
to x2 =
4
2
3p
to x2 = p
4
In Exercises 99–104, find two values of u, 0 … u 6 2p, that satisfy
each equation.
99. sin u =
In Exercises 61–86, use reference angles to find the exact value of
each expression. Do not use a calculator.
17p
b
3
22
2
61. cos 225°
62. sin 300°
63. tan 210°
22
2
64. sec 240°
65. tan 420°
66. tan 405°
103. tan u = - 23
101. sin u = -
100. cos u =
1
2
102. cos u = -
1
2
104. tan u = -
23
3
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514 Chapter 4 Trigonometric Functions
Writing in Mathematics
Preview Exercises
105. If you are given a point on the terminal side of angle u,
explain how to find sin u.
106. Explain why tan 90° is undefined.
107. If cos u 7 0 and tan u 6 0, explain how to find the
quadrant in which u lies.
108. What is a reference angle? Give an example with your
description.
109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.
Exercises 114–116 will help you prepare for the material covered
in the next section. In each exercise, complete the table of
coordinates. Do not use a calculator.
Critical Thinking Exercises
114. y = 12 cos 14x + p2
Chapter
4
0
p
8
p
4
115. y = 4 sin A 2x x
p
3
2p
3
7p
12
13p
12
4p
3
B
5p
6
y
116. y = 3 sin p2 x
x
0
1
3
5
3
1
2
7
3
11
3
3
4
y
After completing this table of coordinates, plot the nine
ordered pairs as points in a rectangular coordinate system.
Then connect the points with a smooth curve.
Mid-Chapter Check Point
What You Know: We learned to use radians to measure
angles: One radian (approximately 57°) is the measure of
the central angle that intercepts an arc equal in length to
the radius of the circle. Using 180° = p radians,
p
b and
we converted degrees to radians a multiply by
180°
180°
b . We defined the six
radians to degrees a multiply by
p
trigonometric functions using coordinates of points
along the unit circle, right triangles, and angles in
standard position. Evaluating trigonometric functions
using reference angles involved connecting a number of
concepts, including finding coterminal and reference
angles, locating special angles, determining the signs of
the trigonometric functions in specific quadrants, and
finding the function values at special angles. Use the
important Study Tip on page 512 as a reference sheet to
help connect these concepts.
In Exercises 1–2, convert each angle in degrees to radians. Express
your answer as a multiple of p.
1. 10°
- p8
y
Make Sense? In Exercises 110–113, determine whether
each statement makes sense or does not make sense, and explain
your reasoning.
110. I’m working with a quadrantal angle u for which sin u is
undefined.
111. This angle u is in a quadrant in which sin u 6 0 and
csc u 7 0.
112. I am given that tan u = 35 , so I can conclude that y = 3 and
x = 5.
113. When I found the exact value of cos 14p
3 , I used a number of
concepts, including coterminal angles, reference angles,
finding the cosine of a special angle, and knowing the
cosine’s sign in various quadrants.
- p4
x
In Exercises 5–7,
a. Find a positive angle less than 360° or 2p that is coterminal
with the given angle.
b. Draw the given angle in standard position.
c. Find the reference angle for the given angle.
11p
19p
6. 7. 510°
3
4
8. Use the point shown on the unit circle to find each of the six
trigonometric functions at t.
5.
y
x2 + y2 = 1
t
t
x
(1, 0)
(
3
4
P 5, 5
)
9. Use the triangle to find each of the
six trigonometric functions of u.
B
2. -105°
6
5
In Exercises 3–4, convert each angle in radians to degrees.
3.
5p
12
4. -
13p
20
u
A
C