Understanding Radian Measure
Many units of measure come from seemingly arbitrary and archaic roots. Some even change over time. The meter, for example was originally intended to be based on the circumference of the earth and now has an amazingly complicated scientific definition! See the resources for further reading. We typically use degrees to measure angles. Exactly what is a degree? A degree is of a complete rotation around a circle. Radians are alternate units used to measure angles in trigonometry. Just as it sounds, a radian is based on the radius of a circle. One radian is the angle created by bending the radius length around the arc of a circle. Because a radian is based on an actual part of the circle rather than an arbitrary division, it is a much more natural unit of angle measure for upper level mathematics and will be especially useful when you move on to study calculus.
What if we were to rotate all the way around the circle? Continuing to add radius lengths, we find that it takes a little more than of them to complete the rotation.
But the arc length of a complete rotation is really the circumference! The circumference is equal to the times the length of the radius. is approximately , so the circumference is a little more than lengths. Or, in terms of radian measure, a complete rotation is .
With this as our starting point, we can find the radian measure of other angles easily. Half of a rotation, or , must therefore be , and must be one-half pi. Complete the table below:
Angle in Degrees Angle in Radians
Because is half of , half of one-half is one-fourth . is one-third of a right angle, so multiplying gives:
and because is twice as large as :
Here is the completed table:
Angle in Degrees Angle in Radians
The last value was found by adding the radian measures of and :
There is a formula to help you convert between radians and degrees that you may already have discovered and we will discuss shortly, however, most angles that you will commonly use can be found easily from the values in this table, so learning them based on the circumference should help increase your comfort level with radians greatly. For example, most students find it easy to remember and . is over and is over . If you know these angles, you can find any of the special angles that have reference angles of and because they will all have the same denominators. The same is true of multiples of pi over and pi over .
”Count”ing in Radians
Do you remember as a child watching the Count on Sesame Street? He would count objects like apples, “one apple, two apples, three apples” and then laugh fiendishly as lightning and thunder erupted around him. Well, to be successful with radian measure, you need to learn to count all over again using radians instead of apples. Let’s start counting right angles, which are really .
“one over , two over (really just ), three over (a ha, ha, ha, ha!!!), four over (which is really )”
Figure 2.1
rotations expressed in radian measure.
You just covered all the angles that are multiples of in one rotation.
Here is the drawing for degree angles:
Figure 2.2
degree rotations
Notice that the additional angles in the drawing all have reference angles of and their radian measures are all multiples of . Complete the following radian measures by counting in multiples of and :
Figure 2.3
degree reference angles
Figure 2.4
degree reference angles
Figure 2.5
degree reference angle radian measure through one rotation.
Figure 2.6
degree reference angle radian measure through one rotation
Notice that all of the angles with degree reference angles are multiples of , and all of those with degree reference angles are multiples of . If you can learn to count in these terms, rather than constantly having to convert back to degrees, it will help you to be effective dealing with most radian measures that you will encounter.
For other examples there is a formula. Remember that:
If you divide both sides of this equality by you will uncover the formula for easy conversion:
so
If we have a degree measure and wish to convert it to radians, then manipulating the equation above gives:
Example 1
Convert to degree measure
Well, if you followed the last section, you should recognize that this angle is a multiple of (or ), so there are , in this angle, .
Here is what it would look like using the formula:
Example 2
Convert to radian measure.Leave the answer in terms of .
Using the formula:
and reducing to lowest terms gives:
However, you could also realize that is . Since is , then is , , or . Make it negative and you have the answer, .
Example 3
Express in degree measure.
Note: Sometimes students have trouble remembering if it is or . It might be helpful to remember that radian measure is almost always expressed in terms of . If you want to convert from radians to degrees, you want the to cancel out when you multiply, so it must be in the denominator.
Radians, Degrees, and a Calculator
Most scientific and graphing calculators have a [MODE] setting that will allow you to either convert between the two, or to find approximations for trig functions using either measure. It is important that if you are using your calculator to estimate a trig function that you know which mode you are using. Look at the following screen:
If you entered this expecting to find the sine of degrees you would realize based on the last chapter that something is wrong because it should be . In fact, as you may have suspected, the calculator is interpreting this as radians. In this case, changing the mode to degrees and recalculating we give the expected result.
Scientific calculators will usually have a letter display that shows either DEG or RAD to tell you which mode you are in. Always check before calculating a trig ratio!!
Example 4
Find the tangent of
First of all, shame on you if you are using a calculator to find this answer! You should know this one! is a quadrant angle with a reference angle of . The tangent of is , and because tangent is negative in quadrant II, the answer is . To verify this on your calculator, make sure the mode is set to Radians, and evaluate the .
Example 5
Find the value of to four decimal places.
Again, you should know the exact value based on your previous work. has a reference angle of and the sign of is . Because is in the quadrant, the cosine is positive and so the exact answer is . Using the calculator gives:
Which, when rounded, is . You can verify that it is indeed a very good approximation of our exact answer using your calculator as well.
Example 6
Convert 1 radian to degree measure.
Many students get so used to using in radian measure that they incorrectly think that means . While it is more convenient and common to express radian measure in terms of , don’t loose sight of the fact that is actually a number! It specifies an angle created by a rotation of approximately lengths. So is a rotation created by an arc that is only a single radius in length. Look back at Figure 1.1. What would you estimate the degree measure of this angle to be? It is certainly acute and appears similar to a angle. To find a closer approximation, we will need the formula and a calculator.
So would be . Using any scientific or graphing calculator will give a reasonable approximation for this degree measure, approximately .
Example 7
Find the radian measure of an acute angle with a .
First of all, it is important to understand that your calculator will most likely not give you radian measure in terms of , but a decimal approximation instead. In this case you need to use the inverse sine function.
This answer may not look at all familiar, but may sound familiar to you. It is an approximation of . So, as you may know, this is
really a angle. Sure enough, evaluating will show that the calculator is giving its best approximation of the radian measure.
If it bothers you that they are not exactly the same, good, it should! Remember that is only an approximation of , so we are already starting off with some rounding error.
Lesson Summary
Angles can be measure in degrees or radians. A radian is the angle defined by an arc length equal to the radius length bent around the circle. One complete rotation around a circle, or is equal to . To convert from degrees to radians you use the following formula:
To convert from radians to degrees the formula becomes:
Much like learning a foreign language where you have to memorize vocabulary to be successful, it will be very helpful for you to understand and be able to communicate in radian measure if you become familiar with the radian measures of the quadrant angles and special angles
Further Reading
http://www.mel.nist.gov/div821/museum/timeline.htm
http://en.wikipedia.org/wiki/Degree_(angle)
http://www.joyofpi.com/
Review Questions
The following picture is a sign for a store that sells cheese.
Figure 2.7
Estimate the degree measure of the angle of the circle that is missing.
Convert that measure to radians.
What is the radian measure of the part of the cheese that remains?
Convert the following degree measures to radians. Give exact answers in terms of , not decimal approximations.
Convert the following radian measures to degrees:
The drawing shows all the quadrant angles as well as those with reference angles of and . On the inner circle, label all angles with their radian measure in terms of and on the outer circle, label all the angles with their degree measure.
Using a calculator, find the approximate degree measure (to the nearest tenth) of each angle expressed in radians.
Gina wanted to calculate the cosine of and got the following answer on her calculator:
Write the correct answer.
Explain what she did wrong.
Fortunately, Kylie saw her answer and told her that it was obviously incorrect.
Complete the following chart. Write your answers in simplest radical form.
Review Answers
Answer may vary, but seems reasonable.
Based on the answer in part a., the ration masure would be
Again, based on part a.,
The correct answer is
Her calculator was is the wrong mode and she calculated the sine of radians.
Applications of Radian Measure
Learning Objectives
A student will be able to:
Solve problems involving angles of rotation using radian measure.
Solve problems by calculating the length of an arc.
Solve problems by calculating the area of a sector.
Approximate the length of a chord given the central angle and radius.
Introduction
In this lesson students will apply radian measure to various problem-solving contexts involving rotations.
Rotations
Example 1
The hands of a clock show 11:20. Express the obtuse angle formed by the hour and minute hands in radian measure to the nearest tenth of a radian.
The following diagram shows the location of the hands at the specified time.
Because there are increments on a clock, the angle between each hour marking on the clock is (or ). So, the angle between the and the is (or ). Because the rotation from to is one-third of a complete rotation, it seems reasonable to assume that the hour hand is moving continuously and has therefore moved one-third of the distance between the and the . So, , and the total measure of the angle is therefore . Using a calculator to approximate the angle would give:
To the nearest tenth of a radian it is .
Length of Arc
The length of an arc on a circle depends on both the angle of rotation and the radius length of the circle. If you recall from the last lesson, we defined a radian as the length of the arc the measure of an angle in radians is defined as the length of the arc cut off by one radius length, so that a half-rotation is , or a little more than lengths around the circle. What if the radius is ? The length of the half-circle arc would be lengths, or in length.
This results in a formula that can be used to calculate the length of any arc.
where is the length of the arc, is the radius, and is the measure of the angle in radians.
Solving this equation for will give us a formula for finding the radian measure given the arc length and the radius length:
Example 2
The free-throw line on an NCAA basketball court is wide. In international competition, it is only about . How much longer is the half circle above the free-throw line on the NCAA court?
Arc Length Calculations:
So the answer is approximately
This is approximately , or about longer.
Example 3
Two connected gears are rotating. The smaller gear has a radius of and the larger gear’s radius is . What is the angle through which the larger gear has rotated when the smaller gear has made one complete rotation?
Because the blue gear performs one complete rotation, the length of the arc traveled is:
So, an arc length on the larger circle would form an angle as follows:
So the angle is approximately .
Area of a Sector
One of the most common geometric formulas is the area of a circle:
In terms of angle rotation, this is the area created by .
A half-circle, or rotation would create a section, or sector of the circle equal to half the area or:
So an angle of would define an area of a sector equal to:
From this we can determine the area of the sector created by any angle, , to be:
Example 4
Crops are often grown using a technique called center pivot irrigation that results in circular shaped fields.
Figure 2.8
Here is a satellite image taken over fields in Kansas that use this type of irrigation system. You can read more about this at: http://en.wikipedia.org/wiki/Center_pivot_irrigation
Figure 2.9
If the irrigation pipe is in length, what is the area that can be irrigated after a rotation of ?
Using the formula:
The area is approximately .
Length of a Chord
You may recall from your Geometry studies that a chord is a segment that begins and ends on a circle.
is a chord in the circle.
We can calculate the length of any chord if we know the angle measure and the length of the radius. Because each endpoint of the chord is on the circle, the distance from the center to and is the same as the radius length.
Next, if we bisect angle, the angle bisector must be perpendicular to the chord and bisect it (we will leave the proof of this to your Geometry class!). This forms a right triangle.
We can now use a simple sine ratio to find half the chord, called here, and double the result to find the length of the chord.
So the length of the chord is:
Example 5
Find the length of the chord of a circle with radius and a central angle of . Approximate your answer to the nearest .
It’s always a good problem solving technique to estimate the answer first. A thought process for estimating the measure might look something like this:
The angle is slightly more than a , or . is slightly more than lengths. One and a half radii would be , so we might expect the answer to be a little more than . Let’s see how the actual answer compares.
We must first convert the angle measure to radians:
Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle.
(Make sure your calculator is in radians!!!)
Multiply this result by 2.
So, the length of the arc is approximately . This seems very reasonable based on our estimate.
Further Reading
http://en.wikipedia.org/wiki/Basketball_court
http://en.wikipedia.org/wiki/Center_pivot_irrigation
http://www.colorado.gov/dpa/doit/archives/history/symbemb.htm#Flag
CK-12 Trigonometry Page 7