Now the circumference of a circle is an arc length. And the ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of ?.
Thus the radian measure is based on ratios -- numbers -- that are actually found in the circle. The radian measure is a real number that indicates the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle ?.
radian measure of the central angle.
At that central angle, the arc is four fifths of the radius.
a) At a central angle of 2.35 radians, what ratio has the arc to the radius?
Answer. That number is the ratio. The arc is 2.35 times the radius.
b) In which quadrant of the circle does 2.35 radians fall?
| Answer. Since ? = 3.14, then | ?
2 | is half of that: 1.57. | 3?
2 | = 3.14 + 1.57 |
= 4.71.
An angle of 2.35 radians, then, is greater than 1.57 but less that 3.14. It falls in the second quadrant.
s = r?
If the radius is 10 cm, and the central angle is 2.35 radians, then how long is the arc?
Answer. We let the definition of theta
become a formula for finding s :
Therefore,
s = 10 × 2.35 = 23.5 cm
Because of the simplicity of that formula, radian measure is used exclusively in theoretical mathematics.
Once again: The radian measure is a real number x. And in the unit circle -- r = 1 -- the length of the arc s is that real number.
s = r? = 1· x = x.
It is here that the term trigonometric "function" has its full meaning. For, corresponding to each real number x -- each radian measure, each arc -- there is a unique value of sin x, of cos x, and so on. The definition of a functionis satisfied. Thus, radian measure can be identified as the length x of an arc of the unit circle. Therefore when we draw the graph of y = sin x we can imagine the unit circle rolled out in both directions onto the x-axis, thus labeling the x-axis.
Because radian measure can be identified as an arc, the inverse trigonometric functions have their names. "arcsin" is the arc -- the radian measure -- whose sine is a certain number.
In the unit circle, the vertical side AB is sin x.
| One of the main theorems in calculus concerns the ratio | sin x
x | for |
very small values of x. And we can see that when the point A is very close to C -- that is, when the central angle AOC is very, very small -- then the opposite side AB will be virtually indistinguishable from the arc length AC. That is,
| sin x |  | x |
| |
sin x
x | | 1. |
An angle of 1 radian
Note that an angle of 1 radian is a central angle whose subtending arc is equal in length to the radius.
That is often cited as the definition of radian measure. Yet it remains to be proved that an arc equal to the radius in one circle, will subtend the same central angle as an arc equal to the radius in another circle. The main theorem cannot be avoided. (Moreover, although we can define an "angle of 1 radian," does such an angle exist? Can we know it? Is it possible to construct it?)
Problem 1.
| a) At a central angle of | ?
5 | , approximately what ratio has the arc to the |
a) radius? Take
?
3.
| The radian measure | ?
5 | is that ratio | . Taking ? 3, then the |
arc is approximately three fifths of the radius.
b) If the radius is 15 cm, approximately how long is the arc?
| s = r?15· | 3
5 | = 9 cm |
Problem 2. In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take
?3.
| a) | ?
4 | | s = r?4· | 3
4 | = 3 cm |
| b) | ?
6 | | s = r?4· | 3
6 | = 4· ½ = 2 cm |
| c) | 3?
2 | | s = r?4· | 3· 3
2 | = 4· | 9
2 | = 2· 9 = 18 cm |
d) 2?. (Here, the arc length is the entire circumference!)
| s = r? = 4· 2?4· 6 = 24 cm |
Problem 3. In which quadrant of the circle does each angle, measured in radians, fall?
Therefore, ? = 2 falls in the second quadrant.
| b) ? = 5 | | 5 radians are more than | 3?
2 | but less than 2?. |
Therefore, ? = 5 falls in the fourth quadrant.
| c) ? = 14 | | 14 radians are more than 2 revolutions, but slightly |
less than 2¼: 6.28 + 6.28 = 12.56.
Therefore, ? = 14 falls in the first quadrant.
Proof of the theorem
In any circles, the same ratio of arc length to radius
determines a unique central angle that the arcs subtend.
If, proportionally,
then
?1 = ?2.
For,
implies, on dividing each side by 2?,
But 2?r is the circumference of each circle. And each circumference is an "arc" that subtends four right angles at the center.
Arcs, moreover, have the same ratio to one another as the central angles they subtend. Therefore,
and
Hence,
This implies
?1 = ?2.
Therefore, the same ratio of arc length to radius determines a unique central angle that the arcs subtend. Which is what we wanted to prove.
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That ratio of the circumference of a circle C to the radius r -- 2
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