Review Questions
The following image shows a clock in Curitiba, Paraná, Brasil.
Figure 2.10
What is the angle between each number of the clock expressed in: exact radian measure in terms of ?
to the nearest tenth of a radian?
in degree measure?
Estimate the measure of the angle between the hands at the time shown in: to the nearest whole degree
in radian measure in terms of
The following picture is a window of a building on the campus of Princeton University in Princeton, New Jersey.
Figure 2.11
What is the exact radian measure in terms of between two consecutive circular dots on the small circle in the center of the window?
If the radius of this circle is about , what is the length of the arc between the centers of each consecutive dot? Round your answer to the nearest .
Now look at the next larger circle in the window.
Find the exact radian measure in terms of between two consecutive dots in this window.
The radius of the glass portion of this window is approximately . Calculate an estimate of the length of the highlighted chord to the nearest . Explain the reasoning behind your solution.
The state championship game is to be held at Ray Diaz Memorial Arena. The seating forms a perfect circle around the court. The principal of Archimedes High School is sent the following diagram showing the seating allotted to the students at her school.
the students from Archimedes.
general admission.
the press and officials.
It is from the center of the court to the beginning of the stands and from the center to the end. Calculate the approximate number of square feet each of the following groups has been granted:
This is an image of the state flag of Colorado
Figure 2.12
The detailed description of the proportions of the flag can be found at: http://www.colorado.gov/dpa/doit/archives/history/symbemb.htm#Flag
It turns out that the diameter of the gold circle is the total height of the flag (the same width as the yellow stripe) and the outer diameter of the red circle is of the total height of the flag. The angle formed by the missing portion of the red band is . In a flag that is tall, what is the area of the red portion of the flag to the nearest square inch?
Review Answers
. Answers may vary, anything above and less than is reasonable.
Again, answers may vary
Let’s assume, to simplify, that the chord stretches to the center of each of the dots. We need to find the measure of the central angle of the circle that connects those two dots.
Since there are dots, this angle is . The length of the chord then is:
The chord is approximately .
Each section is . The area of one section of the stands is therefore the area of the outer sector minus the area of the inner sector: The students have sections or
There are general admission sections or
There is only one press and officials section or
The area of each section is approximately .
There are many difference approaches to the problem. Here is one possibility:
First, calculate the area of the red ring as if it went completely around the circle:
Next, calculate the area of the total sector that would form the opening of the “”
Then, calculate the area of the yellow sector and subtract it from the previous answer.
Finally, subtract this answer from the first area calculated.
The area is approximately
Circular Functions of Real Numbers
Learning Objectives
A student will be able to:
Identify the basic trigonometric ratios as continuous functions of the angle of rotation around the origin.
Identify the domain and range of the six basic trigonometric functions.
Identify the radian and degree measure, as well as the coordinates of points on the unit circle for the quadrant angles, and those with reference angles of and .
Introduction
In this lesson students will view the trigonometric ratios of angles of rotation around the coordinate grid as a continuous, circular function. The connection will be made between how the ratios change as the angle of rotation increases or decreases, and how the graph of the function depicts that change.
y = sin(x), The Sine Graph
By now, you have become very familiar with the specific values of sine, cosine, and tangents for certain angles of rotation around the coordinate grid. In mathematics, we can often learn a lot by looking at how one quantity changes as we consistently vary another. In this case, what will happen to the value of, let’s say, the sine of the angle as we gradually rotate around the coordinate grid. We would be looking at the sine value as a function of the angle of rotation around the coordinate grid. We refer to any such function as a circular function, because they can be defined using the unit circle. First of all, you may recall from earlier sections that the sine of an angle in standard position in the coordinate grid is the ratio of , where is the coordinate of any point on the angle and is the distance from the origin to that point.
Because the ratios are the same for a given angle, regardless of the length of the radius , we can use the unit circle to make things a little more convenient.
The denominator is now , so we have the simpler expression . The advantage to this is that we can use the coordinate of the point on the unit circle to trace the value of through a complete rotation. Imagine if we start at and then rotate counter-clockwise through gradually increasing angles. Since the coordinate is the sine value, watch the height of the point as you rotate.
Through Quadrant I that height gets larger, starting at , increasing quickly at first, then slower until the angle reaches , at which point, the height is at its maximum value, .
As you rotate into the third quadrant, the change in the height now reverses itself and starts to decrease towards .
When you start to rotate into the third and fourth quadrants, the length of the segment increases, but this time in a negative direction, growing to at and heading back toward at .
After one complete rotation, even though the angle continues to increase, the sine values will simply repeat themselves. The same would have been true if we chose to rotate clockwise to investigate negative angles, and this explains why the sine function is periodic. The period is or , because that is the angle measure required before the sine of the angle will simply repeat the previous sequence of values.
Let’s translate this circular motion into a graph of the sine value vs. the angle of rotation. The following sequence of pictures demonstrates the connection. As the angle of rotation increases, watch the coordinate of the point on the angle as it traces horizontally. Ignore the values along the horizontal axis at this point as they just relative. What is important is that you make the connection between the circular rotation and the change in the height of the point.
Notice that once we rotate around once, the point traces back over the same values again. The red curve that you see is one period of a sine “wave”. If you would like to see this happen in “real time”, look at one of the links in the readings section or just do a search for Java applets and “sine” online and you will find many excellent demonstrations.
Let’s look at some specific values so we can graph the sine function more precisely. Since we already know what happens in between, you can draw a fairly accurate sketch by plotting the points for the quadrant angles
The value of goes from to to to and back to . Graphed along a horizontal axis showing , it would look like this:
Filling in the gaps in between and allowing for multiple rotations as well as negative angles results in the graph of where is any angle of rotation(usually expressed in radians):
As we have already mentioned, has a period of . You should also note that the values never go above or below , so the range of
a sine wave is . Because we can continue to spin around the circle forever, there is no restriction on the angle , so the domain of is all reals.
y = cos(x), The Cosine Graph
In chapter 1, you learned that sine and cosine are very closely related. The cosine of an angle is the same as the sine of the complementary angle. So, it should not surprise you that sine and cosine waves are very similar in that they are both periodic with a period of , a range from to , and a domain of all real angles.
The cosine of an angle is the ratio of , so in the unit circle, the cosine is the coordinate of the point of rotation. If we trace the coordinate through a rotation, you will notice the change in the distance is similar to , but starts in a different place. The coordinate of a angle is and the coordinate for is , so the cosine value is decreasing from to through the quadrant.
Here is a similar sequence of rotations to the one we used for sine. This time compare the coordinate of the point of rotation with the height of the point as it traces along the horizontal.
Plotting the quadrant angles and filling in the in-between values shows the graph of
The graph of has a period of . Just like , the values never “escape” from the unit circle, so they stay between and . The range of a cosine wave is also . And also just like the sine function, there is no restriction on the angle of rotation, so the domain of is all reals.
y = tan(x), The Tangent Graph
The graph of the tangent ratio as a function of the angle of rotation presents a few complications. First of all, the domain is no longer all real angles. As you may remember there are some angles ( and , for example) for which the tangent is not defined. As we will see in this section, the range of is actually all real numbers.
The measurement of each of the six trig functions can be found by using a single segment from the unit circle, however, the remaining functions are not as obvious as sine and cosine. The name of the tangent function comes from the tangent line, which is a line that is perpendicular to the radius of a circle at a point on the circle so that the line touches the circle at exactly one, and only one, point. So, to create the tangent segment, first we draw a tangent line perpendicular to the axis.
If we extend angle through the unit circle so that it intersects with the tangent line, the tangent function is defined as the length of the red segment.
The dashed segment is because it is the radius of the unit circle. Recall that the tangent of is , so we can verify that this segment is indeed the tangent by using similar triangles.
So, as we increase the angle of rotation, think about how this segment changes. When the angle is , the segment has no length. As we begin to rotate through the first quadrant, it will increase, very slowly at first.
But, you can see very soon that the value increases past one. As the angle gets closer to , the segment will need to stretch quite high in order to intersect the extension of the angle and it will grow at a faster and faster rate.
As we get very close to the axis that the segment gets infinitely large, until when the angle really hits , at which point the extension of the angle and the tangent line will actually be parallel and therefore never meet!
This means there is no definition for the length of the tangent segment, or as it may be helpful to think of it, the tangent segment is infinitely large.
Before continuing, let’s take a look at this portion of the graph through the first quadrant. The tangent starts at , for a angle, then increases slowly at first. That increase gets much steeper and as we approach a rotation.
Again, just a small break in the axis on these graphs will make it more clear that these two concepts to not lie side-by-side on the same coordinate grid.
In fact as we get infinitely close to , the tangent value increases without bound, until when we actually reach , at which point the tangent is undefined. A line that a graph gets infinitely close to without touching is called an asymptote. So the tangent function has an asymptote at .
As we rotate past , now the intersection of the extension of the angle and the tangent line is actually below the axis. This fits nicely with what we know about the tangent for a quadrant angle being negative. It will be first be very, very negative, but as the angle rotates, the segment gets shorter, reaches , then crosses back into the positive numbers as the angle enters the quadrant.
The segment will again get infinitely large as it approaches . After being undefined at , the angle crosses into the quadrant and once again changes from being infinitely negative, to approaching zero as we complete a full rotation.
So, this motion graphed over several rotations would look like this:
Notice that the axis is measured in radians (not in terms of ). Our asymptotes occur every , starting at . The period of the graph is therefore . The domain is all reals except for the “holes” at and the range is all real numbers.
The Three Reciprocal Functions: cot(x), csc(x), and sec(x)
Cotangent
Cotangent is the reciprocal of tangent, so it makes sense to generate the circular function for cotangent by drawing the tangent line at a point on the axis and extending the angle, instead of the axis.
We can verify that this is the case again by using similar triangles. Because the purported cotangent segment is parallel to the base of the yellow triangle, then angle is in the opposite corner and the triangles are indeed similar, even though their positions are reversed.
So,
Now that we have established the cotangent segment, think about how this segment changes as we rotate around the coordinate grid starting at . First of all, at itself, the cotangent is undefined because the segment is parallel to the ray of the angle . As we begin to increase the angle of rotation, the segment will be extremely large and begin to get smaller as we approach , very quickly at first, but then slowing down as it gets closer to length at .
After passing , the segment will again start to lengthen, but this time it will be in the negative direction, increasing slowly at first, then getting infinitely large in the negative direction until , at which point it is again undefined.
After passing this point, the periodic behavior kicks in and the function now repeats the same sequence of values as we rotate from , back to .
Tracing this motion on the graph over several rotations gives:
Remember that cotangent and tangent are reciprocals of each other, so any point at which the tangent was equal to , the cotangent will be undefined and any point at which the tangent was undefined, the cotangent is equal to .
You might also notice that the graphs consistently intersect at and . These are the angles that have reference angles, which always have tangents and cotangents equal to or . It makes sense that and are the only values for which a function and it’s reciprocal are the same. Keep this in mind as we look at cosecant and secant compared to their reciprocals of sine and cosine.
The cotangent function has a domain of all real angles except multiples of The range is all real numbers.
Cosecant
There are many ways possible to find the cosecant segment. One approach is to look at the right triangle formed by the cotangent segment and use the Pythagorean Theorem to generate the cosecant.
From the original triangle in the unit circle,
Since is cosecant, then the cosecant must be the same as side .
Tracing the length of this segment, it is undefined at , infinitely large for very small angles, decreasing to at and then increasing infinitely until it is undefined at . The process repeats from to , however, the segment starts infinitely negative, increases to at before approaching an infinitely negative length.
The period of the function is therefore with a domain of all real angles except multiples of . The range is all real numbers greater than or less than .
The graph then would look as follows:
Here is the graph of as well:
Notice again the reciprocal relationships at and the asymptotes. Also look at the intersection points of the grap
hs at and . Many students are reminded of parabolas when they look at the half period of the cosecant graph. While they are similar in that they each have a local minimum or maximum and they begin and end in the same direction the comparisons end there and they shouldn’t be referred to as parabolic. The mathematics that defines the values, and therefore shape, of the graph is completely different from the quadratic function of a parabola.
Secant
Much like the relationship between sine and cosine, secant and cosecant share many similarities. The segment used to generate is shown below:
You will be asked to demonstrate this in the exercises section. This segment is for , then grows through the first quadrant, and is undefined at . It is infinitely negative shrinking down to through the quadrant, before lengthening back towards infinite negativity and is undefined at . Translating this motion to a graph of gives us:
CK-12 Trigonometry Page 8
