This week we will be starting a new Trigonometric Identity Chapter which will involve deriving some key angle formulas and applying them to find exact values. The early part of the chapter will involve a lot of memorizing and applying some relatively long identities and you will really need to be sure you have mastered much of the material from Chapter 3 (working with the unit circle, definition of all the trig functions and their reciprocals etc.) To start off the week I want to see who is following this blog and who is actively reading their textbook and involve some students in actually creating the weekly assignment sheet and responding to some questions.
Three Key Things
- I had mentioned to some D-period students that since they had completed their after test assignment more thoroughly than others, they may get an extra credit opportunity. That opportunity is now if you go to the Ch. 6 draft assignment sheet and replace the ???? with the written objectives (find in your textbook) or names of the identities (find in textbook). Each student to fill in an objective or key identity will be exempt from part of a future homework. Access the draft assignment sheet by clicking the link below and it should let you make edits. https://docs.google.com/document/d/1vUohN90Jrh5DBIpLOPXEy6FgyR6yl_DvtYm4eR5uC7g/edit
- Secondly, for anyone who wants to answer any of the questions below by posting a comment to this blog, you may also receive reduced homework in the near future if you answer accurately. [D- or C-Period: Only one response per question, check comments before replying]
- The assignment sheet is a "working assignment sheet" some of the problem sets may change but not by much and we are actually starting out with 6.2 which is not on the assignment sheet but will be announced in class. Our first quiz for this Chapter will probably be Friday Day3.
1) Describe what "dead reckoning" is for ship navigation and explain how you think this may have changed given more current technologies.
2) How is a ship navigation angle reading convention different than the angle convention we use in Trigonometry class.
3) Describe what a "knot" in terms of sailing navigation and provide a little history by doing some online research as to where the measurement of "knots" came from.
4) What is a sextant and how is it used?
5) What are the similarities and differences between the cos (A+/-B) and sin (A+/-B) Identities.
Good navigating has its rewards. If you are posting, be sure to state the question # you are replying to.
Hi, I'm answering #1.
ReplyDeleteDead reckoning is determining a ship's position based on their previous one while taking into account speed, direction, and the period of time elapsed. It is an estimate and can be quite inaccurate. Today, there is no use for dead reckoning since ships nowadays use global positioning units and other technologies which locate a ship's exact position. Thus dead reckoning is used as a last resort.
Best Regards,
Kyle Hassan
Hi Mr. Steppen,
ReplyDeleteI am answering number four. A sextant is an instrument that is used to measure the angle between any two visible objects. In order to use the sextant, the telescope must be focused on the horizon. Then the sextant must be aimed at a celestial body (star). Then one must bring the celestial body down to the horizon by moving the arm along the arc and then clamp the arm. Then one must use the micrometer knob to make small adjustments while gently swaying the instrument slightly from side to side until the heavenly body just brushes the horizon. After this happens the exact time must be noted down in the order of seconds, minutes, then hours, and the celestial body that was observed, as well as the measure of the angle.
Thanks,
Jerry Haas
I'm going to answer #3. A knot is a unit of speed that means one nautical mile per hour. Abbreviated "kn", a knot is equal to about 1.151 miles per hour. The origin, according to my dad who was a sailor, was they would find the speed of ships by throwing a chip log overboard which was a weighted piece of wood and there was a rope tied to it. There would be knots tied about every 47 feet. (I had to google that.) An 30 second hand glass would time how long it took for knots to pass through a sailor's hands and then the results would be reported to the captain.
ReplyDeleteThanks,
Shannon Gallagher
Hi! I'm going to take a whack at #5.
ReplyDeleteFirst off, the four identities are:
Cosine Difference Identity: cos(A-B) = cosAcosB + sinAsinB
Cosine Sum Identity: cos(A+B) = cosAcosB - sinAsinB
Sine Difference Identity: sin(A-B) = sinAcosB - cosAsinB
Sine Sum Identity: sin(A+B) = sinAcosB + cosAsinB
The obvious similarity is that both are A+/-B identities. One more similarity between the cos(A+/-B) and sin(A+/-B) identities is that they are set up in the same way. Also, both only use the two trigonometric functions sine and cosine. Another similarity is that the two identities (or technically four identities) use only the variables given (A and B) and don't introduce any new ones.
The obvious difference is that two are sine identities and two are cosine identities. Another difference between the identities is that the cosine sum identity subtracts(cos(A-B)=cosAcosB+sinAsinB) and the difference identity adds (cos(A+B)=cosAcosB-sinAsinB) while the sine sum identity adds(sin(A-B)=sinAcosB-cosAsinB) and the difference identity subtracts (sin(A+B)=sinAcosB+cosAsinB). Another difference is that under both cosine identities, cosA, cosB, sinA and sinB are multiplied by their matching function with the other variable (cosAcosB and sinAsinB) while in the sine identities, they are multiplied by the other function with the other variable (sinAcosB and cosAsinB).
These are the only similarities and differences I could think of, although I am sure there are more. Hopefully this about covers it though...
Thanks!
Isabel Macomber
Oh, now I get it. It just takes the lure of extra credit to get some comments. Kyle, Jerry, Shannon, and Isabel can skip the HW assignment 6.2 (from today) since they were the first posters.
ReplyDeleteIll answer #2
ReplyDeleteShips don't use the unit circle because that wouldn't work on earth, as it wouldn't cover everything. Ships use longitude and latitude to navigate. Longitude are lines which go from north to south pole and go 90 degrees north from the equator, or 0 degrees, and 90 degrees south from the prime meridian. Latitude are lines which go around the earth and go 180 degrees west and 180 degrees east from the prime meridian, or 0 degrees. This is further divided to minutes and seconds, allowing ships to get a very accurate position of themselves. Longitude and latitude are use(d) in gps, dead reckoning, and with sextants.
bowdan is alek, crazy random account needed to be made
ReplyDeletehello Mr. Steppen i'll do #3, like Shannon has explained, knots is a unit of measurement when dealing with speed. where 1 knot = 1.151 mph, and where a nautical mile is 6,076.12 ft. With knots this was for sailing ships to find out how fast they were going at anytime. They would have a riel of rope that had knots on it every 50 feet, and at the end was a chipped piece of wood. They'd throw the chipped piece of wood into the water, start their hourglass, and see how many knots would be pulled into the water after a certain amount of time had passed. with the conversions and formula we can find the speed:
ReplyDelete50 ft / 0.5 sec X 100ft / min X 1NM / 6076ft X 60 min / hr = 1NM / hr
Putting in the right numbers that deal with your situation will help you find out the speed and distance you'll travel. For example, if 10 knots are pulled into the water in 30 seconds, the ship will be traveling at 10 knots.
Thanks,
Dennis Turano
Good explanations about the knots, I always found that interesting.
ReplyDeleteAlek, who I thought for a second was maybe just a curious observer of SteppenintoTrig, has partially covered #2. I was thinking of what we consider the "starting angle". Due North does make sense in navigation, but in our class we are more partial to 3:00 as our zero angle and 4:30 as a shout out to the Spin Doctors. "What time is it" - http://www.spindoctors.com/music/pocket-full-kryptonite
I'll try #5. The sextant is an instrument for measuring angular distances used especially in navigation. The sextant specifically measures the angle of elevation of stars or other celestial bodies. The sextant uses an optical principle that states that when a ray of light is reflected from two successive mirrors then the angle between the first and last direction of the ray of light is twice the angle between the mirrors. To use the sextant, the telescope (which composes one part of the device) must be pointed at the horizon.
ReplyDelete