Comparing it with the cosine graph:
The period is , the range is the same as , and the domain is all real angles except multiples of
Lesson Summary
The six trigonometric functions defined by the ratios in a right triangle can be placed in the context of the coordinate grid by thinking of them in terms of a point rotating around a circle centered at the origin with a radius of one. This circle is called the unit circle. The sine of the angle of rotation is the coordinate of the point, the cosine of the angle is the coordinate, and the tangent is . The values of the other three ratios; cotangent, cosecant, and secant can also be found in terms of their reciprocal relationships, but all of these values can be constructed geometrically as various segments around the angle of rotation on the unit circle. Instead of finding isolated values, we can look at each ratio as a function of the angle of rotation. These are called circular functions. Here are the domains and ranges of the six circular trigonometric functions.
Function Domain Range
all reals
all reals
all reals
all reals
Further Reading
http://www.mathnstuff.com/math/spoken/here/2class/330/unit.htm
http://mathdemos.gcsu.edu/mathdemos/family_of_functions/trig_gallery.html
http://mathforum.org/library/topics/trig/branch.html
Review Questions
Show that side A in this drawing is equal to
In Chapter 1, you learned that . Use the drawing and results from question 1 to demonstrate this identity.
This diagram shows a unit circle with all the angles that have reference angles of , , and , as well as the quadrant angles. Label the coordinates of all points on the unit circle. On the smallest circle, label the angles in degrees, and on the middle circle, label the angles in radians.
Draw and label the line segments in the following drawing that represent the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent)
Which of the following shows functions that are both increasing as increases from to ? and
and
and
and
Which of the following statements are true as increases from to ? approaches
gets infinitely large
gets infinitely small
Review Answers
Use similar triangles:
So:
Using the Pythagorean theorem then, .
b
d
Linear and Angular Velocity
Learning Objectives
A student will be able to:
Calculate linear velocity.
Calculate angular velocity.
Apply the calculation of linear velocity to real-world situations.
Apply the calculation of angular velocity to real-world situations.
Introduction
In this lesson students will review the formula and calculation of simple velocity and use them to investigate some practical applications. Then students will then discover that the velocity of an object traveling around a circle can be calculated in terms of the angle of rotation and apply that relationship in practical situations.
Linear Velocity v = s/t
Linear Velocity
Table of Interesting Velocities
This table lists some common velocities from some of the fastest moving particles and objects, to some of the slowest. Notice in each case that the units used to measure velocity can be used to remember the formula for velocity. In each case the velocity, or speed (we’ll let your physics teacher clarify the difference between the two) is expressed in terms of a distance divided by a unit of time.
In symbolic form we will use to represent distance(sometimes referred to as position) Replacing for velocity and for time, results in the formula:
You may remember from some of your earlier math courses, that this is simply the distance formula (distance = rate, or velocity in this case, times the time) solved for the value of the velocity. The distance traveled is equal to the speed at which you travel, multiplied by the time for which you have been going at that speed.
Solve the equation for by dividing both sides by .
The cancels on the right side and the result is the velocity formula.
If we are using this formula to calculate the velocity of someone or something, that object must be moving at a constant speed. If not, the velocity that we calculate is an average velocity for the entire time period.
Example 1
A toy racecar is traveling around an oval track (at a constant rate) that measures in length. It takes the car to complete one lap. Find the speed of the car in feet per second.
Using the formula, replace the values given for distance and time:
Example 2
Lois finishes her cross-country race in 0:21:14 . What is her average velocity for the race? Express this velocity in miles per hour.
In this example, it is reasonable to assume that she is not going to run at the same rate of speed for the entire race, and therefore our calculation will result in an average velocity. She may start out at the beginning of the race quickly to establish a position before settling into a reasonable pace. Parts of the course may be up or downhill causing her velocity to change, and there may even be a sprint for the finish line.
First, we need to take her time and change it to a single unit. Let’s express it in minutes only. There are in a minute so is of a minute. Her total time in minutes is:
Now we can use the formula:
Because there are in every hour, multiply this result by to express the speed in miles per hour.
Her average speed was approximately .
Example 3
Gauss and Newton are riding bicycles toward each other at constant rates of and , respectively. They start apart. Meanwhile, at the same time, a fly flying starts at Gauss and travels towards Newton. When the fly reaches Newton, it turns around and flies back to Gauss. The fly continues flying back and forth between the two riders until they collide and crush the fly. What is the total distance the fly has traveled?
This problem, though perhaps not too realistic, is often used to teach students problem solving techniques. Can you think of any ways in which we have to simplify the situation in order to do any calculations? One significant simplification has to do with the fly changing directions. As we have learned, in order to calculate simple velocity, we must assume that the object in question is traveling at a constant rate. It is actually physically impossible for the fly to instantaneously change directions and maintain the exact same speed. In order to simplify the problem, let’s say that the fly is somehow able to maintain a constant rate of speed.
Even after making that assumption, many students struggle with this problem. It is tempting to use the distance formula to begin to measure each leg of the fly’s journey. The problem quickly becomes overwhelming when viewed in this manner. The trick to solving it is to change your point of view. Even though we have titled this section “linear velocity,” it might be more properly named “constant velocity.” As long as the fly is traveling at the same rate of speed, the distance it travels is related to that speed and the time, not the direction that the fly travels. So the fact that the fly is flying back and forth can be ignored in the solving of the problem.
Look at the following diagram of the problem:
How long does it take for the riders to collide? If Gauss rides for , he will have gone exactly . Similarly, Newton will travel in one hour. Because the total distance is , they will collide in exactly one hour! So, the fly will also travel for one hour at a constant rate of . The fly will cover a total distance of , whether it flies in a straight line, circles, or back and forth!
Angular Velocity
Angular Velocity
In the last example, we introduced the idea that the direction of motion does not affect the calculation of velocity. What about objects that are traveling on a circular path? Do you remember playi
ng on a merry-go-round when you were younger (or maybe you don’t want to admit it was last week!)?
Figure 2.13
If two people are riding on the outer edge, their velocities should be the same. But, what if one person is close to the center and the other person is on the edge? They are on the same object, but their speed is actually not the same.
Look at the following drawing.
Imagine the point on the larger circle is the person on the edge of the merry-go-round and the point on the smaller circle is the person towards the middle. If the merry-go-round spins exactly once, then both individuals will also make one complete revolution in the same amount of time. However, it is obvious that the person in the center did not travel nearly as far. The circumference (and of course the radius) of that circle is much smaller and therefore the person who traveled a greater distance in the same amount of time is actually traveling faster, even though they are on the same object. So the person on the edge has a greater linear velocity. If you have ever actually ridden on a merry-go-round, you know this already because it is much more fun to be on the edge than in the center! But, there is something about the two individuals traveling around that is the same. They will both cover the same rotation in the same period of time. This type of speed, measuring the angle of rotation over a given amount of time is called the angular velocity.
The formula for angular velocity is:
is the last letter in the Greek alphabet, omega, and is commonly used as the symbol for angular velocity. is the angle of rotation expressed in radian measure, and is the time to complete the rotation.
In this drawing, is exactly one radian, or the length of the radius bent around the circle. If it took point A exactly to rotate through the angle, the angular velocity of would be:
In order to know the linear speed of the particle, we would have to know the actual distance, that is, the length of the radius. Let’s say that the radius is .
If linear velocity is:
then,
or
If the angle were not exactly , then the distance traveled by the point on the circle is the length of the arc. You may recall from an earlier section that the formula for arc length is:
or, the radius length times the measure of the angle in radians. Substituting into the formula for linear velocity gives:
Pull the out in front:
Notice the formula for angular velocity! Substituting gives the following relationship between linear and angular velocity.
so the linear velocity is equal to the radius times the angular velocity.
Remember that in a unit circle, the radius is , so it turns out that the linear velocity is the same as the angular velocity.
In this case the actual distance traveled around the circle is the same for a given unit of time as the angle of rotation measured in radians.
Example 4
Lindsay and Megan are riding on a Merry-go-round. Megan is standing from the center and Lindsay is riding on the outside edge from the center. It takes them to complete a rotation. Calculate the linear and angular velocity of each girl.
We are told that it takes to complete a rotation. A complete rotation is the same as . So the angular velocity is:
per second, which is slightly more than (about ), radian per second. Because both girls cover the same angle of rotation in the same amount of time, their angular speed is the same. In this case they rotate through approximately of the circle every second.
As we discussed previously, their linear velocities are different. Using the formula, Megan’s linear velocity is:
Lindsay’s linear velocity is:
Lesson Summary
The linear velocity of an object is defined as the distance traveled divided by the time, or . The angular velocity is a measure of the angle of rotation through which a point rotates around a circular path and is found by dividing the angle of rotation by the time, or . If you know the angular velocity of an object, you can find its linear velocity by multiplying the angular velocity and the radius length of the rotation circle:
Further Reading
http://en.wikipedia.org/wiki/Speed_of_light
http://hypertextbook.com/facts/2001/InnaSokolyanskaya1.shtml
http://en.wikipedia.org/wiki/Fingernails
http://en.wikipedia.org/wiki/Cheetah
http://www.pbs.org/odyssey/voice/20000503_vos_transcript.html
Review Questions
Figure 2.14
Suppose the radius of the dial of an electric meter on a house is . How fast is a point on the outside edge of the dial moving if it completes a revolution in ?
Find the angular velocity of a point on the dial.
Suppose the person inside the house from question turns on the air-conditioner, turns on all the lights in the house, boots up several computers, turns on a big-screen tv, makes a piece of toast, and heats up his coffee in the microwave. At that moment, it takes only for the dial to complete a rotation. Calculate the velocity of the point on the outside of the dial.
Calculate the angular velocity.
Doris and Lois go for a ride on a carousel. Doris rides on one of the outside horses and Lois rides on one of the smaller horses near the center. Lois’ horse is from the center of the carousel, and Doris’ horse is farther away from the center than Lois’. When the carousel starts, it takes them to complete a rotation. Calculate the velocity of each girl.
Calculate the angular velocity of the horses on the carousel.
The large hadron collider near Geneva, Switerland began operation in 2008 and is designed to perform experiments that physicists hope will provide important information about the underlying structure of the universe. The LHC is circular with a circumference of approximately . Protons will be accelerated to a speed that is very close to the speed of light How long does it take a proton to make a complete rotation around the collider?
What is the approximate (to the nearest meter per second) angular speed of a proton traveling around the collider?
Approximately how many times would a proton travel around the collider in one full second?
Sources:
http://en.wikipedia.org/wiki/Large_Hadron_Collider
http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/lhc-vital-statistics.htm
Review Answers
Lois: , Doris:
or hundred-thousandths of a second
about rotations in just a second!!!!
Graphing Sine and Cosine Functions
Learning Objectives
A student will be able to:
Identify periodic functions.
Identify the basic graphs of and .
Calculate the amplitude of a sine or cosine wave.
Calculate the period of a sine or cosine wave.
Calculate the frequency of a sine or cosine wave.
Graph transformations of sine and cosine waves involving changes in amplitude and period (frequency).
Introduction
In this lesson students will generalize their knowledge of the basic trigonometric ratios to investigate the functional behavior of sine and cosine values. Because these values are repetitive, an understanding of the behavior of periodic functions will be developed. Students will learn the basic characteristics of periodic functions including period, amplitude, and frequency. The students will be expected to be able to identify these characteristics from an equation, or use them to create a graph.
Periodic Functions
“I just need to get back into a routine.”
“I am just stuck in this routine and I can’t break out of it.”
You have probably heard people make, or perhaps even made yourself, similar comments about events in their lives. The fact that human behavior seems to be repetitive can be both a good thing, and a bad one. Think about all the things you do during a typical day that are part of a routine that repeats over and over again. The alarm clock, your breakfast, the radio or television schedule, the traffic lights on the way to schoo
l, and your class schedule tend to be the same every day, and while we sometimes complain about the drudgery, many would say that we also need this repetition to be healthy. If you look outside of our own behavior to the world around us we see repetitive phenomena in the seasons, sunlight, and weather. If we started tracking the time of the sunrise in a particular city (ignoring the changes of daylight savings time) on January 1, we would see it gradually getting earlier towards summer and becoming later into the fall and winter. This cycle would repeat itself all over again if we continued to track it for a second year. Situations that behave in this manner are called periodic. If this behavior is measurable and we graph the change s over time , the resulting graph of is called a periodic function. The period is the distance we must travel on the axis before the function repeats itself.
CK-12 Trigonometry Page 9