2
( Hint: Mimic Example 3.18 using (sin A + sin B) + (sin C − sin ( A + B + C)) . ) 11. cos A + cos ( B − C) = 2 sin B sin C
12. sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C
( Hints: Group sin 2 B and sin 2 C together, use the double-angle formula for sin 2 A, use Exercise
11. )
a − b
sin A − sin B
13.
=
a + b
sin A + sin B
s ( s − a)
( s − b) ( s − c)
14. cos 1 A
and sin 1 A
, where s
( a
2
=
=
= 1
+ b + c)
bc
2
bc
2
( Hint: Use the Law of Cosines to show that 2 bc (1 + cos A) = 4 s ( s − a) . )
15. 1 (sin A
( A
2
+ sin B) ≤ sin 12
+ B)
( Hint: Show that sin 1 ( A
(sin A
2
+ B) − 12
+ sin B) ≥ 0 . )
16. In Example 3.20, which angles A, B, C give the maximum value of cos A + cos B + cos C ?
4Though it does not matter for this exercise, none of the angles in these formulas are measured in degrees. We
will discuss their unit of measurement in Chapter 4.
4 Radian Measure
4.1 Radians and Degrees
So far we have been using degrees as our unit of measurement for angles. However, there is
another way of measuring angles that is often more convenient. The idea is simple: associate
a central angle of a circle with the arc that it intercepts.
Consider a circle of radius r > 0, as in Figure 4.1.1. In geometry you learned that the
circumference C of the circle is C = 2 π r, where π = 3.14159265....
AB = 1 C
r
AB
C
4
= π 2
= 12 = π r
AB = C = 2 π r
B
90◦
180◦
360◦
A
A
B
A
O
O
O
B
(a) θ = 90◦
(b) θ = 180◦
(c) θ = 360◦
Figure 4.1.1
Angle θ and intercepted arc AB on circle of circumference C = 2 πr
In Figure 4.1.1 we see that a central angle of 90◦ cuts off an arc of length π r, a central
2
angle of 180◦ cuts off an arc of length π r, and a central angle of 360◦ cuts off an arc of length
2 π r, which is the same as the circumference of the circle. So associating the central angle
with its intercepted arc, we could say, for example, that
360◦
“equals”
2 π r
(or 2 π ‘radiuses’).
The radius r was arbitrary, but the 2 π in front of it stays the same. So instead of using the
awkward “radiuses” or “radii”, we use the term radians:
360◦ = 2 π radians
(4.1)
The above relation gives us any easy way to convert between degrees and radians:
π
Degrees to radians:
x degrees
=
· x radians
(4.2)
180
180
Radians to degrees:
x radians
=
· x
degrees
(4.3)
π
87
88
Chapter 4 • Radian Measure
§4.1
Formula (4.2) follows by dividing both sides of equation (4.1) by 360, so that 1◦ = 2 π
360 = π
180
radians, then multiplying both sides by x. Formula (4.3) is similarly derived by dividing
both sides of equation (4.1) by 2 π then multiplying both sides by x.
The statement θ = 2 π radians is usually abbreviated as θ = 2 π rad, or just θ = 2 π when it is clear that we are using radians. When an angle is given as some multiple of π, you can
assume that the units being used are radians.
Example 4.1
Convert 18◦ to radians.
Solution: Using the conversion formula (4.2) for degrees to radians, we get
π
π
18◦ =
· 18 =
rad .
180
10
Example 4.2
Convert π radians to degrees.
9
Solution: Using the conversion formula (4.3) for radians to degrees, we get
π
180
π
rad =
·
= 20◦ .
9
π
9
Table 4.1
Commonly used angles in radians
Degrees
Radians
Degrees
Radians
Degrees
Radians
Degrees
Radians
π
3 π
0◦
0
90◦
180◦
π
270◦
2
2
π
2 π
7 π
5 π
30◦
120◦
210◦
300◦
6
3
6
3
π
3 π
5 π
7 π
45◦
135◦
225◦
315◦
4
4
4
4
π
5 π
4 π
11 π
60◦
150◦
240◦
330◦
3
6
3
6
Table 4.1 shows the conversion between degrees and radians for some
r
common angles. Using the conversion formula (4.3) for radians to degrees,
θ
we see that
180
O
r
1 radian =
degrees ≈ 57.3◦ .
π
Formally, a radian is defined as the central angle in a circle of radius
θ = 1 radian
r which intercepts an arc of length r, as in Figure 4.1.2. This definition
Figure 4.1.2
does not depend on the choice of r (imagine resizing Figure 4.1.2).
Radians and Degrees • Section 4.1
89
One reason why radians are used is that the scale is smaller than for degrees. One revolu-
tion in radians is 2 π ≈ 6.283185307, which is much smaller than 360, the number of degrees
in one revolution. The smaller scale makes the graphs of trigonometric functions (which we
will discuss in Chapter 5) have similar scales for the horizontal and vertical axes. Another
reason is that often in physical applications the variables being used are in terms of arc
length, which makes radians a natural choice.
The default mode in most scientific calculators is to use degrees for entering angles. On
✄
many calculators there is a button labeled DRG for switching between degree mode (D),
✂
✁
radian mode (R), and gradian mode (G).1 On some graphing calculators, such as the the TI-
✄
83, there is a MODE button for changing between degrees and radians. Make sure that your
✂
✁
calculator is in the correct angle mode before entering angles, or your answers will likely be
way off. For example,
sin 4◦ =
0.0698 ,
sin (4 rad) = −0.7568 ,
so the values are not only off in magnitude, but do not even have the same sign. Using your
✄
✄
✄
calculator’s sin−1 , cos−1 , and tan−1 buttons in radian mode will of course give you the angle
✂
✁ ✂
✁
✂
✁
as a decimal, not an expression in terms of π.
You should also be aware that the math functions in many computer programming lan-
guages use radians, so you would have to write your own angle conversions.2
Exercises
For Exercises 1-5, convert the given angle to radians.
1. 4◦
2. 15◦
3. 130◦
4. 275◦
5. −108◦
For Exercises 6-10, convert the given angle to degrees.
π
11 π
29 π
6. 4 rad
7.
rad
8.
rad
9.
rad
10. 35 rad
5
9
30
11. Put your calculator in radian mode and take the cosine of 0. Whatever the answer is, take its
cosine. Then take the cosine of the new answer. Keep repeating this. On most calculators after
about 50-60 iterations you should start to see the same answer repeating. What is that number?
Try starting with a number different from 0. Do you get the same answer repeating after roughly
the same number of iterations as before? Try the same procedure in degree mode, starting with 0◦.
Does the same thing happen? If so, does it take fewer iterations for the answer to start repeating
than in radian mode, or more?
1A gradian is defined as 1 of a circle, i.e. there are 400 gradians in one revolution. Compared to the more
400
common 360◦ in one revolution, gradians appear to be easier to work with, since a right angle is 100 gradians
(thus making integer multiples of a right angle easier to remember). Outside of a few specialized areas (e.g.
artillery calculations), gradians are nevertheless not widely used today.
2One exception is Octave, which has functions cosd(), sind(), tand() that take angles in degrees as parameters,
in addition to the usual cos(), sin(), tan() functions which use radians.
90
Chapter 4 • Radian Measure
§4.2
4.2 Arc Length
In Section 4.1 we saw that one revolution has a radian measure of 2 π rad. Note that 2 π is
the ratio of the circumference (i.e. total arc length) C of a circle to its radius r:
2 π r
C
total arc length
Radian measure of 1 revolution = 2 π =
=
=
r
r
radius
Clearly, that ratio is independent of r. In general, the radian measure of an angle is the
ratio of the arc length cut off by the corresponding central angle in a circle to the radius of
the circle, independent of the radius.
To see this, recall our formal definition of a radian: the central angle in a circle of radius r
which intercepts an arc of length r. So suppose that we have a circle of radius r and we place
a central angle with radian measure 1 on top of another central angle with radian measure
1, as in Figure 4.2.1(a). Clearly, the combined central angle of the two angles has radian
measure 1 + 1 = 2, and the combined arc length is r + r = 2 r.
r
r/2
2
r
1
r/2
1 1
1/2
r
r
(a) 2 radians
(b) 1 radian
2
Figure 4.2.1
Radian measure and arc length
Now suppose that we cut the angle with radian measure 1 in half, as in Figure 4.2.1(b).
Clearly, this cuts the arc length r in half as well. Thus, we see that
Angle = 1 radian
⇒
arc length = r ,
Angle = 2 radians ⇒
arc length = 2 r ,
Angle = 1 radian
r ,
2
⇒
arc length = 12
and in general, for any θ ≥ 0,
Angle = θ radians ⇒
arc length = θ r ,
so that
arc length
θ =
.
radius
Arc Length • Section 4.2
91
Intuitively, it is obvious that shrinking or magnifying a circle preserves the measure of a
central angle even as the radius changes. The above discussion says more, namely that the
ratio of the length s of an intercepted arc to the radius r is preserved, precisely because that
ratio is the measure of the central angle in radians (see Figure 4.2.2).
s = r′ θ
s = rθ
θ
θ
O
r′
O
r
(a) Angle θ, radius r
(b) Angle θ, radius r′
Figure 4.2.2
Circles with the same central angle, different radii
We thus get a simple formula for the length of an arc:
In a circle of radius r, let s be the length of an arc intercepted by a central angle with
radian measure θ ≥ 0. Then the arc length s is:
s = r θ
(4.4)
Example 4.3
In a circle of radius r = 2 cm, what is the length s of the arc intercepted by a central angle of measure
θ = 1.2 rad ?
Solution: Using formula (4.4), we get:
s = r θ = (2)(1.2) = 2.4 cm
Example 4.4
In a circle of radius r = 10 ft, what is the length s of the arc intercepted by a central angle of measure
θ = 41◦ ?
Solution: Using formula (4.4) blindly with θ = 41◦, we would get s = r θ = (10)(41) = 410 ft. But this impossible, since a circle of radius 10 ft has a circumference of only 2 π (10) ≈ 62.83 ft! Our error was
in using the angle θ measured in degrees, not radians. So first convert θ = 41◦ to radians, then use
s = r θ:
π
θ = 41◦ =
· 41 = 0.716 rad
⇒
s = r θ = (10)(0.716) = 7.16 ft
180
Note that since the arc length s and radius r are usually given in the same units, radian
measure is really unitless, since you can think of the units canceling in the ratio s , which is
r
just θ. This is another reason why radians are so widely used.
92
Chapter 4 • Radian Measure
§4.2
Example 4.5
A central angle in a circle of radius 5 m cuts off an arc of length 2 m. What is the measure of the
angle in radians? What is the measure in degrees?
Solution: Letting r = 5 and s = 2 in formula (4.4), we get:
s
2
θ =
=
= 0.4 rad
r
5
In degrees, the angle is:
180
θ = 0.4 rad =
· 0.4 = 22.92◦
π
For central angles θ > 2 π rad, i.e. θ > 360◦, it may not be clear what is meant by the inter-
cepted arc, since the angle is larger than one revolution and hence “wraps around” the circle
more than once. We will take the approach that such an arc consists of the full circumference
plus any additional arc length determined by the angle. In other words, formula (4.4) is still
valid for angles θ > 2 π rad.
What about negative angles? In this case using s = r θ would mean that the arc length is
negative, which violates the usual concept of length. So we will adopt the convention of only
using nonnegative central angles when discussing arc length.
Example 4.6
A rope is fastened to a wall in two places 8 ft apart at the same height. A
A
cylindrical container with a radius of 2 ft is pushed away from the wall as
far as it can go while being held in by the rope, as in Figure 4.2.3 which
B
shows the top view. If the center of the container is 3 feet away from the
4
point on the wall midway between the ends of the rope, what is the length
2
L of the rope?
θ
D
C
3
E
Solution: We see that, by symmetry, the total length of the rope is L =
2 ( AB + BC). Also, notice that △ ADE is a right triangle, so the hypotenuse
4
has length AE = DE 2 + D A 2 = 32 + 42 = 5 ft, by the Pythagorean The-
orem. Now since AB is tangent to the circular container, we know that
∠ ABE is a right angle. So by the Pythagorean Theorem we have
Figure 4.2.3
AB =
AE 2 − BE 2 =
52 − 22 =
21 ft.
By formula (4.4) the arc BC has length BE · θ, where θ = ∠ BEC is the
supplement of ∠ AED + ∠ AEB. So since
4
BE
2
tan ∠ AED =
⇒ ∠ AED = 53.1◦ and