A Problem From Afar

My Dear Dear students from last year! It's been a while and I needed to know that you haven't stopped thinking and started memorizing random facts. I miss you all terribly and to show how much I miss you I shall pose a small question to you that should surely look like a "mole hill" to you when compared to your prior achievements. 

Is there a geometric proof of the algebraic statement below?

a² – b² = (a+b)(a-b)

If so, share your suggestion or proof by commenting to this post and remember, you'll have to communicate a visual idea through words because it'll be hard to share images when commenting. Goodluck!

Any Questions?

You can use this post for questions you may have regarding the topics we have covered in both geometry and trigonometry.

Homework due 05/20/09

So, given the radius of a circle:

1. Can you find the area of a sector if you know the central angle that the minor arc makes?

2. Can you find the area of a sector if you know the arc length that the sector intercepts?

(Note: The sector of a circle is like a pizza slice, in case you were wondering what a sector is!)

Homework due 05/15/09

Find the area of the triangle below. (Hint: You will need to find at least one angle, and I would suggest you employ the use of an altitude line.)

Homework due 05/12/09

You have the worksheet!

Homework due 05/11/09

1. On February 10, 1990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0.1 feet. Assuming the next high tide is at exactly 12 noon and that the height of the water is given by a sine or cosine curve, find a formula for the water level in Boston as a function of time, measured in hours since midnight.

2. Of course, there's something wrong with the assumption in the problem above that the next high tide is exactly at noon. If so, the high tide would always be at noon or midnight, instead of progressing slowly through the day, as in fact it does. The interval between successive high tides actually averages about 12 hours 24 minutes. Using this, give a more accurate formula for the height of the water as a function of time.

Simple Harmonic Motion

So you obviously know that you have to plot the tangent functions graph using your unit circles and a table of values. Once you are done with that, you need to establish improvements/refinements to your method of getting precise measurements to craft a function that does a good job of modeling the "simple harmonic motion" of a mass suspended at the end of a spring. Think about:

(a) Number of oscillations in a given time period.

(b) The period (time taken for one oscillation).

(c) The lowest height, the greatest height and the equilibrium position.

(d) How you can utilize all the members of your group in an efficient way.

There may be other considerations and you should certainly discuss them using the blog. On Friday, without a single minute to waste, you must be ready to commence with your function modeling exercise. The end goal is that you come up with a function that fairly accurately models the motion of the suspended mass.

A "Transforming" Experience

So folks, you have some handouts/explorations for recollecting the topic of function transformations. See how much you can recall and grasp. Write down any observations you make. Try taking the test and seeing where you stand. You can use this post to discuss things. Just remember, if you find yourself doing the "if you see this then do this" routine then STOP! Don't do that! I'd rather you not know something then know it that way.

Homework due 05/05/09

You have the worksheet that I gave you in class. Remember, if it takes you longer then that time is worth spending now. Take more time now and you will naturally be able to sort this stuff out with more ease later. Check the conference folder early in the morning or very late at night for the answers to the worksheet.

Homework due 05/01/09

Folks, your assignment for Friday is the handout I gave you in class. If you misplace it or forget it in school you should download it from the AGT conference folder in First Class. If some of you felt overwhelmed by today's class then I suggest looking over the notes and the whole deal with radian measure, degree measure, negative angles and what they mean and what we mean by angles larger than 360 degrees or 2Pi radians. Click here for an image that can help you with the questions about quadrants.

Below are some things to remember for the homework assignment:

In question 1 you should refer to your unit circles and/or your trig tables. Remember that the sign (- or +) for sines, cosines and tangents becomes very important here. If angles are negative, then figure out which positive angles they are "co-terminal" with to figure out the sines, cosines and tangents. If the angles are larger then 360 degrees or much smaller than 360 degrees (such as - 405 degrees) then do the same, i.e. figure out which angle it is co-terminal with and then find the corresponding function values from your unit circle. Below is a link to an image that I think will help you understand what is meant by these new types of angles we are witnessing:
http://media.wiley.com/Lux/65/10665.nfg003.jpg

In question 2 you should note that I have asked for two solutions to the equations. In other words, there are two angles between 0 and 360 degrees that correspond to those particular lengths of lines that each equation depicts. Look at your unit circles and this will make more sense. Make sure to give the angles in both radians and degrees as the question demands. Remember, no calculators here as well.

Question 3 should be a straightforward question that you need no special hints for.

Homework due 04/28/09

Re-read the Jamshed al-Kashi article by Glen Van Brummelen and this time focus on al-Kashi's approximation of Pi. Re-try his method in your homework journals, grappling with all the arguments and seeing how he geometrically derives his formula. It will require your prior knowledge of geometry. Be ready to discuss your exploration in class.

Homework due 04/27/09

Your homework is the worksheet handed out to you in class and in case you misplace it or leave it behind in school, you can download it from the class conference folder, AGT - Kerai. Please do the work in your homework journals and show all your work.

Quiz on Friday, April 24th

You will be quizzed on whatever we have covered in trigonometry and also on our work with circles. You can use this post to compare answers for the questions you did in class involving right triangles.

Assignment due 04/20/09

This assignment is to be done on seperate sheets of paper as it will be collected on Monday.

1. Using only the formulas we’ve derived so far and the computations of sines of angles we have done, determine the value of sin(3°). (Use your algebraic skills at first and then use your calculators to get a numerical answer.)

2. Read the article on Jamshed al-Kashi.

3. Go back to where the formula involving sin(3 theta) occurs in the article and then attempt problem 4 of this assignment.

4. Using the sine and cosine addition and subtraction identities, prove the formula .

5. Use fixed-point iteration, as outlined in the article, to get a value for sin(1°) to about 6 decimal places of accuracy.

Sine Half-Angle Formula Discussion Post

Hint #1:
Identify which line segments on the diagram are equal to sin(alpha/2) and cos(alpha/2), and note how sin(alpha) and cos(alpha) show up in the diagram.

Hint #2:
What is the value of angle ECB? Are there similar triangle possibilities here?

Hint #3:
CE=CD-AB or AB=CD-CE

Hint #4:
Get an expression for cos(alpha) from triangle ECB. Every term in your expression should be convertible to sines or cosines of alpha and (alpha/2).

Hint #5:
From here it's just algebra. Solve for sin(alpha/2), remembering that you can convert a cosine to a sine using the pythagorean theorem.

Sine Addition Law Discussion Post

Hint #1:
Identify which line segments on the diagram are equal to sin(beta), cos(beta), and the quantity we want, sin(alpha+beta).

Hint #2:
Can you think of a ratio that equals sin(alpha) and one that equals cos(alpha)?

Hint #3:
What is angle CDF equal to ?... are there any similar triangles in the diagram? Which ones?

Hint#4:
See if you can determine values for the lengths FC, FD, EC, and OE in terms of the sines and cosines of alpha and beta.

Homework due 04/14/09

So, now that you have your wonderful trig-table (zij) you can write in the sines, cosines, and tangent values of the special angles +36 degrees, their supplements, and then tick mark all the other sine, cosine and tangent values that you could possibly find using the formulas that you have been given. Remember, that if you can find the sine of a new angle then you can surely use that to find one of yet another new angle. What you CAN find, you can use.

When doing this exhaustive checking off of angles, think of what rational angles sines, cosines and tangents can be found for and what would be necessary in order to be able to find values for all angles.

Note: If you heard what Jack uttered today, then he has already clued you in to the second part of what I am asking you to consider.

Homework due 04/14/09

Consider our table of values that we have started to generate for the sines and cosines of certain angles. So far we have 0, 30, 36, 45, 60, 90 and we can surely find more based on these. However, we then have all the other angles (infinitely many to be precise) for which we do not know sines and cosines and are concerned about how we could generate these values using geometric approaches.

You have been given a formula sheet that consists of relationships that the Muslim astronomers and their Indian and Greek predecessors were well aware of. Although we may not use these without the necessary derivations that we will undertake soon, we can get an idea of the possibilities of angles for which sines and cosines could be determined based on these formulas.

So, list as many angles between 0 and 360 degrees that we can find sines and cosines for based on the formulas you have been given. Also, ponder the first two formulas and determine whether they are just stating what you had already observed before or are they completely new formulas for you. We will discuss this further in class.

Homework due 04/13/09

Geometric Solution to the Sine of 36 Degrees

Okay folks, you have a neat challenge that awaits you on the geometric solution to the sine of 36 degrees. Please use this blog to post comments if you have questions. Remember, this will require you to use all your knowledge of geometry and don't be surprised if a quadratic equation pops up somewhere. You can use your calculators for the computations (to avoid spending hours doing arithmetic as the 10th century astronomers did) but only round off quantities at the very end. In fact, don't round off anything until you get your final answer. Work in your homework journals and show all your work.

Then, read the article on Islamic Astronomy by Owen Gingerich that I handed out at the end of class. This reading will provide you with all the necessary context for the work we are conducting.

(Now please do the reading or I'll be forced to give you another "reading quiz" and then let the dangling question of whether it'll be counted or not be much cause for emotional instability!)

Recommended Practice Questions (continued from class)

Folks, these are the exercises from class. Please make sure to do them if you wish to understand how the theorems pertaining to circles are used. Remember, you will be quizzed on this next week (and it will not be open book).

Page 365-366; #3, 8, 9, 10 and 12
Page 368-369; #4, 9, 13 and 23
Page 371; #8 and 9.