14 ## Arc LengthThe definition of radian measure
IT IS CONVENTIONAL to let the letter 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 π.
Since D = 2
or,
That ratio -- 2π -- of the circumference of a circle to the radius, is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles.
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
Proportionally, if and only if θ1 = θ2. We will prove this theorem below.
At that central angle, the arc is four fifths of the radius.
Example 2. An angle of .75 radians means that the arc is three fourths of the radius. Example 3. In a circle whose radius is 10 cm, a central angle θ intercepts an arc of 8 cm. a) What is the radian measure of that angle?
b) At that same central angle θ, what is the arc length if the radius is 5 cm?
Example 4. a) At a central angle of 2.35 radians, what ratio has the arc to the radius?
b) In which quadrant of the circle does 2.35 radians fall?
An angle of 2.35 radians, then, is greater than 1.57 but less that 3.14. It falls in the second quadrant.
become a formula for finding
Therefore,
Because of the simplicity of that formula, radian measure is used exclusively in theoretical mathematics. The unit circle Since in any circle the same ratio of arc to radius determines a unique central angle, then for theoretical work we often use the unit circle, which is a circle of radius 1: In the unit circle, the radian measure
We can identify radian measure, then, as the length Moreover, when we draw the graph of 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 opposite side AB
One of the main theorems in calculus concerns the ratio
for very small values of
An angle of 1 radian An angle of 1 radian refers to 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 if an arc is equal to the radius in one circle, it will subtend the same central angle as an arc equal to the radius in another circle. We cannot avoid the main theorem. In addition, although it is possible to define an "angle of 1 radian," does such an angle actually exist? Is it possible to draw one -- a curved line equal to a straight line? Or is that but another example of fantasy mathematics? See First Principles of Euclid's Elements, Commentary on the Definitions; see in particular that a definition asserts only how a word or a name will be used. It does not assert that what has been defined exists. Problem 1.
a) the arc to the radius? Take π3.
then the arc is approximately three fifths of the radius. b) If the radius is 15 cm, approximately how long is the arc?
Problem 2. In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take π3.
d) 2π. (Here, the arc length is the entire circumference!)
Problem 3. In which quadrant of the circle does each angle, measured in radians, fall? (See the figure above.)
but less than π. (See the figure above.) Therefore, θ = 2 falls in the second quadrant.
but less than 2π. (See the figure above.) Therefore, θ = 5 falls in the fourth quadrant.
but slightly less than 2¼: 6.28 + 6.28 = 12.56. (See the figure above.) Therefore, θ = 14 falls in the first quadrant. Proof of the theorem In any circles the same ratio of arc length to radius Proportionally, if and only if θ1 = θ2. For, if and only if Now 2π
But in the same circle, arcs have the same ratio to one another as the central angles they subtend. (Theorem 16.) Therefore, and Therefore, according to line (1), if and only if θ1 = θ2. 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. Next Topic: Analytic Trigonometry and the Unit Circle Please make a donation to keep TheMathPage online. Copyright © 2016 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |