http://www.managementhelp.org/prsn_prd/prob_slv.htm
http://office.microsoft.com/en-us/onenote/HA101686341033.aspx
http://en.wikipedia.org/wiki/Image:Circle-trig6.svg [diagram]
http://www.ies.co.jp/math/java/trig/sixtrigfn/sixtrigfn.html
Tuesday, March 18, 2008
Performance Task- Letter to LCBC
Dear LCBC
It is an honor that you have given me the opportunity to propose to you my model of the unit circle. Trigonometry is a very important subject to consider, even in every day life. Trigonometry has many applications in the real world, as I will explain in my proposal. For now, I will explain how the trigonometric functions are derived. It all begins with a unit circle. A unit circle is a circle with radius 1. It is from this circle that we can see all 6 trigonometric functions. On this circle, draw a triangle, assuming a 30 degree angle.
We know that the hypotenuse has a length of one. To get the Sine of a triangle, you divide the opposite side by the hypotenuse. Because the hypotenuse is equal to 1, Sine is equal to the opposite side, which is shown by the red line. Cosine is found in a similar way. Cosine is the adjacent side divided by the hypotenuse. Again, since the hypotenuse is equal to 1, Cosine is equal to the adjacent side, as noted by the purple line. Shown below is the unit circle if we add Tangent and Secant. If you draw a line at the point (1,0) and extend it upwards to create a new triangle on the unit circle, it is easy to see where Tangent and Secant are found. First, let's derive Tangent. Tangent is the opposite angle divided by the adjacent one. Because this is a unit circle, we know that the adjacent angle is equal to 1, so the opposite angle is equal to the Tangent function, as shown by the color green. Secant is found by dividing the hypotenuse by the adjacent. We know that the adjacent angle is one, therefore, the hypotenuse is equal to secant, as shown by the color blue.
Now we will build the third and final triangle in order to get Cotangent and Cosecant. This is done by extending a line out from the (0,1) position in the graph, as shown below. Just a note, that in this triangle, we will be using a different angle. Because the angle we have been using we have assumed is equal to 30 degrees, its complimentary angle must be equal to 60 degrees. From here we know that the other angle in the triangle is 30 degrees, which is the angle we will now be using. Cotangent is the reciprocal function of tangent, and is therefore found by dividing the adjacent angle by the opposite angle. From this diagram, we know that the opposite angle is equal to one. The adjacent side is equal Cotangent, as shown in pink.
By dividing the hypotenuse by the opposite side, we are left with Cosecant. And, since the opposite side we know to be equal to one, Cosecant is equal to the hypotenuse. And there we have all six trigonometric functions.
It is extremely important to have a strong grasp of the unit circle, trigonometric functions and what they all mean because trigonometry is very important to modern day society. My unit circle model and description clearly describe how the six trigonometric functions are derived, and during my proposal, I will describe the importance of these functions in modern day, common occupations. Thank you for your time.
Sincerely,
Student Name
It is an honor that you have given me the opportunity to propose to you my model of the unit circle. Trigonometry is a very important subject to consider, even in every day life. Trigonometry has many applications in the real world, as I will explain in my proposal. For now, I will explain how the trigonometric functions are derived. It all begins with a unit circle. A unit circle is a circle with radius 1. It is from this circle that we can see all 6 trigonometric functions. On this circle, draw a triangle, assuming a 30 degree angle.
We know that the hypotenuse has a length of one. To get the Sine of a triangle, you divide the opposite side by the hypotenuse. Because the hypotenuse is equal to 1, Sine is equal to the opposite side, which is shown by the red line. Cosine is found in a similar way. Cosine is the adjacent side divided by the hypotenuse. Again, since the hypotenuse is equal to 1, Cosine is equal to the adjacent side, as noted by the purple line. Shown below is the unit circle if we add Tangent and Secant. If you draw a line at the point (1,0) and extend it upwards to create a new triangle on the unit circle, it is easy to see where Tangent and Secant are found. First, let's derive Tangent. Tangent is the opposite angle divided by the adjacent one. Because this is a unit circle, we know that the adjacent angle is equal to 1, so the opposite angle is equal to the Tangent function, as shown by the color green. Secant is found by dividing the hypotenuse by the adjacent. We know that the adjacent angle is one, therefore, the hypotenuse is equal to secant, as shown by the color blue.
Now we will build the third and final triangle in order to get Cotangent and Cosecant. This is done by extending a line out from the (0,1) position in the graph, as shown below. Just a note, that in this triangle, we will be using a different angle. Because the angle we have been using we have assumed is equal to 30 degrees, its complimentary angle must be equal to 60 degrees. From here we know that the other angle in the triangle is 30 degrees, which is the angle we will now be using. Cotangent is the reciprocal function of tangent, and is therefore found by dividing the adjacent angle by the opposite angle. From this diagram, we know that the opposite angle is equal to one. The adjacent side is equal Cotangent, as shown in pink.
By dividing the hypotenuse by the opposite side, we are left with Cosecant. And, since the opposite side we know to be equal to one, Cosecant is equal to the hypotenuse. And there we have all six trigonometric functions.It is extremely important to have a strong grasp of the unit circle, trigonometric functions and what they all mean because trigonometry is very important to modern day society. My unit circle model and description clearly describe how the six trigonometric functions are derived, and during my proposal, I will describe the importance of these functions in modern day, common occupations. Thank you for your time.
Sincerely,
Student Name
Different Approaches to Problem Solving
Today was our last class before the unit ends. Today we talked about the different approaches to problem solving. First we watched another video, by the same guy who did the video we watched last class. This really got us to think about how there are several different ways to "attack" the same problem. To get out heads thinking, we filled out pros and cons sheets for several different types of approaches to the same problem. We brainstormed several ideas with a partner before joining in a class discussion. For homework, we need to pick 2 approaches to research and compare in depth, using a Venn diagram. At the end of class, some time was set aside for us to ask questions one last time over anything that has happened in the unit before we move on to the performance task.
Trig in Real Life
Today we started off class with a really cheesy video about math applications in real life. This video started a discussion about all the different types of professions that utilize trigonometry. Ms. Mitchell had LOTS of examples to share with us including lumberjack, mathematician, pilot, and carpenter. Throughout the period, the discussion continued while we worked. I really enjoyed the free flow feeling of the classroom. We were able to ask questions to our teacher and peers, and used laptops to search online. After we picked a profession, we had to write down three ways trigonometry affects their occupation. As each student finished, we were given immediate teacher feedback, which was really helpful. Our final draft is due next class. It has to be one page describing the occupation, how the occupation utilizes trig, and what it would be like if trig had not been invented yet. For my profession, I picked Astronomy. I can't wait to get started on my paper!
Recognition and Memorization
Today we had a big class discussion and worked on our memorization skills. Throughout the class discussion, we used OneNote to organize our thoughts. OneNote is an amazing program that helps to organize all your work very efficiently and effectively. Watch this video to see more. After the class discussion, we were given the opportunity to ask any questions before we handed in a copy of our notes. Our notes were meant to show our thought process through the discussion and sample examples and memorization techniques. The most common technique is SOHCAHTOA, but I don't like that one at all. The one the teacher gave us is much easier to remember [Ophie had a hit of acid, and Ophie had a hangover]. The first letter of each word corresponds to the six trig function in order. Ophie had = o/h = Sin. During the class discussion we also talked about how you know when to use which function, which is pretty easy. That is what the quiz was on too, so I think I did pretty well.
Tree Activity
Today in class we were given our first project! At the beginning of the class, we were taught how trig functions are applied inside a right triangle. We had already been introduced to the trig functions, so that part was easy. Next we learned how to problem solve using trig functions. After given several examples to practice, as well as plenty of opportunity to get feedback on these in class problems, we were split up into groups of 3s to go outside. There we were given specific information about a tree, and we needed to find the height. During class, we gathered up all the information, and all of our work is due next class. I never thought Trig could be applied this way before today, but I am glad that we are doing real applications of Trig functions. They are much easier to learn that way.
The Unit Circle
During the last 2 class periods, we discussed the Unit Circle. The Unit Circle is a circle with radius 1, from which all the Trig functions can be derived. In class we started working on graphing Sin and Cos onto the Unit Circle. It was hard at first, but once we began to discuss exactly what we know about a circle with radius one, it was easy. For Sin, for example, you draw a triangle into the Unit Circle. Sin= Opp/Hyp. Because we know the Hyp=1, Opp must equal Sin. Cos is also found in a similar way. For homework, we were to work on trying to derive other Trig functions, and next class we went over each function, its placement in the Unit Circle, and why it is there. An excellent representation of the Unit Circle is found here. The Unit Circle helps us to understand where the Trig Functions come from, as well as a prequel to their importance in Triangles, as well as circles.
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