So far, our unit of measurement for angles and rotations is the degree. We measure an angle in degrees using a protractor:
Why are there 360 degrees in a full rotation? The answer to this lies in history, not in mathematics. It turns out that there is a more convenient way to measure angles and rotations called radian measure. Mathematicians and scientists find it natural to use radian measure to express relationships.
Unlike degree measure, radian measure does not depend on an arbitrary unit.
To measure an angle in radians, we place its vertex at the center of any circle, and think about the length of arc A B, as measured in inches, centimeters, or some other unit of length:
The length of this arc depends on the size of angle A 0 B:
But is also depends on the size of the radius of the circle:
So we cannot simply take the length of this arc as the measure of the angle. But the ratio of the length of the arc to the radius of the circle depends only on the size of the angle:
Definition. The radian measure of an angle is the ratio of the arc it cuts off to the radius of any circle whose center is the vertex of the angle.
This definition reminds us of the definition of the sine of an angle, which is also a ratio, and which does not depend on the particular right triangle that the angle belongs to.
Example 36 What is the radian measure of an angle of 60°?
Answer. We place the vertex of our 60° angle at the center of a circle of radius r, and examine the arc it cuts off. Since 60/360 = 1/6, this arc is 116 of the circumference of the circle.
Numerically, this is approximately 1.0471976, or a little more than 1 radian.
Example 37 What is the radian measure of an angle of 360°?
Example 38 What is the degree measure of an angle of 1 radian?
In Example 38, we have used the very important fact that radian measure is proportional to degree measure. In fact, it is not hard to see that.
There are two simple tests that this measurement passes. First, the bigger the angle, the bigger its radian measure. Second, if we place two angles next to each other (see figure),
the measure of the larger angle they form together is the sum of the measures of the two
original angles 2.
Recall that if r is the radius of a circle, the length of its circumference is given by the formula the ratio of the radian measure of an angle to its degree measure is always
In general, if we are measuring an angle in radians, we do not use any special symbol like the “degree” sign.
Solution. We first express the angle in degrees. If D is the required degree measure, we have
which leads to D = 30°, and we know that sin 30° = 1/2.
Example 40 In a circle of radius 1, what is the length of the arc cut off by a central angle of 2 radians?
Solution 1 (the long and hard way). We saw above that the degree measure of this angle is about 114°. So the arc cut off by this angle is approximately.
Solution 2 (the neat and easy way). In a unit circle (whose radius is 1 unit), the radian measure of a central angle is just the length of the arc it cuts off. This tells us that the required arc is exactly two units long (and gives us an idea of the error we made in using the approximate degree measure in Solution 1).
Example 41 A central angle in a circle of radius 2 units cuts off an arc 5 units long. What is the radian measure of this angle?
Solution. By definition, this radian measure is 5/2.
Radian measure and distance.
Imagine a wheel whose radius is 1 foot. Let this wheel roll, without slipping, along a straight road:
Since the wheel does not slip as it rolls, the distance it rolls, in feet, is just the length of the arc that the angle a cuts off.
Example 42 How far will a wheel of radius 1 foot travel after 1 rotation?
Solution. Because it rotates without slipping, the wheel will travel exactly the length of the circumference of the circle. But if the radius of the circle is r, then the circumference is
Sometimes a car wheel slips as it rolls. This is called a skid, and it happens when there is not enough friction between the wheel and the road (for example when the road is icy or wet). A car wheel can also tum without rolling: sometimes, a car stuck in deep snow will spin its wheels. We assume that neither of these things is happening to our wheel.
Example 43 A wheel of radius 1 foot rotates through 1/2 a rotation. How far will it travel?
Example 44 How far does a wheel of radius 1 foot travel along a line, if it rotates through an angle of 2 radians?
Answer. Two feet.
Example 45 Through how many radians does a circle of radius 1 foot rotate, if it travels 5 feet down a road?
Answer. Five radians.
Example 46 How much does a wheel with radius 1 foot rotate if it travels 1000 feet along a road? Give the answer in radians and also in degrees.
Solution. In radians, this is easy: it has rotated through 1000 radians. In degrees, the answer is more difficult to find. Each full rotation covers
in radian measure
Example 48 Is sin 500 (in radian measure) a positive or a negative number?
between 112 and 3/4 of one rotation. So a rotation of 500 radians will end up in the third quadrant, and its sine is a negative number.
Interlude: How to explain radian measure to your younger brother or sister
When you drive with Mom or Dad in the car, did you ever notice the odometer? That’s the little row of numbers in front of the steering wheel.
It measures the distance covered by the car, in miles. But how does it know this? The odometer cannot read the road signs, telling us how far we’ve come. It must get the information from the car’s wheels. But the car’s wheels can only tell the odometer how much they have turned. The more the car’s wheels tum, the more distance we cover.
The odometer knows how to convert rotations to miles. In geometry, we
But suppose you want to know how much wear the tires have had.
Then we must read the odometer, and figure out how many rotations the tires have made from the distance they traveled. So if the odometer says that the car has traveled 200 feet (we have to convert from miles, again),
Radian measure and calculators
Most calculators, and all scientific calculators, know about radian measure.
You can switch your calculator between “degree mode” and “radian mode” (and sometimes there are still other w”ays to measure angles). But each calculator does this in a different way. It is important that you know how to tell which mode your calculator is working in, and also how to switch from one mode to another.
An important graph
Let us summarize our knowledge of the sine function by drawing its graph.
The integer multiples of rr will give us a convenient scale for the xaxis, since the values of sin x at these points are easy to calculate. For the y-axis, we need only values from -1 to 1, since sin x can only take on these values.
We can draw the graph by looking at a unit circle (drawn on the right below), and recording the height of a point which makes an angle a with the x-axis. Here is what it looks like for a typical acute angle ot.
And in the third and fourth quadrants, the situation is like this:
Two small miracles
We pause here to describe two remarkable relationships, so remarkable that they seem like miracles. An explanation (that is, a mathematical proof) of these miracles is postponed for later.
Miracle 1: The area under the sine curve
Look at the first arch of the curve y = sin x. What can we tell about
further with the approximations, taking more and more triangles which would “fill” the area below the curve. Something like this is in fact done, in calculus.
The result is a small miracle: The area under one arch of the sine curve is exactly 2.
Miracle 2: The tangent to the sine curve
Let us take a point P = (a, sin a) on the curve y = sinx. Let the perpendicular from P meet the x-axis at the point Q. Let us draw the tangent to the curve at point P, and extend it to meet the x -axis at R. It is easy to see that PQ =sin a.
But, by a small miracle, we can also find the length of Q R. lt is just | tan a |, the absolute value of tan a.
Notice that the radian measure of angles, like their degree measure, is additive. That is, if two angles are placed so as to “add up” to a larger angle, the sum of the angles corresponds to the sum of the arcs 4.
Another good thing about radian measure is that it is dimensionless.
That is, it is independent of any unit of measurement. Length, for instance, can be measured in centimeters, inches, or miles, and we get different numbers.
The same is true of area, volume, and many other quantities. But radian measure, like the sine of an angle, is a ratio, and so does not depend on the units used to measure the arc of the circle or its radius. This is another reason why physicists, and other scientists too, like to use radians.
Since the radian measure and the sine of an angle are both dimensionless, we can compare them. For an acute angle a, which is larger,· sin a or the radian measure of a?
Geometry can help us answer this, if the angle is small. In the diagram below, we took a circle of unit radius, and drew a tiny angle A 0 P. Then we made another copy of this angle (back-to-back with the first copy) and labeled it PO B. Then arc AP =arc P B.
Whenever we decide how to measure something, we would like the measure to be additive. Length is additive, as is area and volume. However, a trip to the grocery will quickly confirm that the price of Coca-Cola is not additive.
The price of two 6-ounce bottles is likely to be more than the price of one 12-ounce bottle, because you are paying for
packaging, labeling, shipping, and so on.
But with radian measure we can even prove a bit more. Later on we will see that for a small angle a measured in radians, the ratio sin a I a is very close to 1. For example, for a = 0.1, sin a is more than 99% of a itself.
But this is true only if we use radian measure for x. In degrees, as we have seen, this formula would be terrible.
because it is an irrational number, and our decimal notational system for numbers doesn’t provide us with a good symbol for it6(this is why we use a Greek letter). But it’s even less convenient for English-speaking people to convert miles to kilometers, or pounds to kilograms. So please don’t let this slight inconvenience stop you from using radian measurement.
What does it mean for two numbers to be “close”? For example, 1 and 0.99 are certainly close: their difference is 0.01, a tiny number. But 1000 and 998 are also close.
Their difference is 2, which is a much larger number than 0.01. However, the ratio 998 : 1000 = 0.998 is very close to l. So sometimes we should measure “closeness” by seeing how close the ratio of two numbers is to l. Thinking this way, we would not say ‘that, 0.1 and 0.0001 are close. Although both these numbers are small, and their difference
is small, their ratio is 1000, which is not small. In the diagram it is true that if a is small, not only is sin a also small, but the two numbers are close, since their ratio is close to l.this slight inconvenience stop you from using radian measurement.