Important Note:There are several video clips on this page, but because of a bug in the way my web host does things, those video clips sometimes don't show up. If you can't see any video clips below, be sure to answer "Show All Content" or "Show Non-Secure Content" when your browser software asks for it when this page opens. If your browser doesn't ask that question and you can't see any video clips then try using a different browser. I'll switch my stuff over to a different web host as soon as I have time to do so. For now... I apologize for the confusion.First, a Pre-Test - Do you have the basic math skills that you'll need? If you can't pass this little test then you won't be able to solve the bigger problem. (But if you don't pass it on your first attempt, see me. I'll gladly help you to pass this pre-test if you'll work with me.) Click on the triangle icon to see the video clip.
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Question 1: What is the length of side "c"?
Need help? Use the Law of Cosines which, for "flat" triangles (the kind the the ancient Greek man named Euclid taught about) is... c
^{2} = a^{2} + b^{2} - (2)(a)(b)cos(C). Solve this for c to get....Question 2: What is the value of angle "B"?
Need help? Rewrite the Law of Cosines in terms of side "b" and angle "B". You'll get... b
^{2} = a^{2} + c^{2} - (2)(a)(c)cos(B). Solve this for B to get...OK... Assuming that you passed that little test, now let's learn about Non-Euclidean Geometry.It turns out that before we can calculate the direction we need to go to get to Hawaii, we first need to calculate the distance. We're going to calculate that distance using a modified version of the Law of Cosines that you just got through proving to me that you know how to use. The reason we have to modify the Law of Cosines is because the triangle we're going to solve isn't the normal "flat" triangle (drawn on a flat surface) that you're used to working with. The triangle that we need to solve is a "spherical triangle" which is drawn on the surface of the Earth. The equation of the Law of Cosines for spherical triangles looks a little different than the one you learned back when you studied Euclidean Geometry. However, you use the equation for exactly the same purpose that you just got through using the one you're familiar with (in the preceding problems). First, let's use an orange to make sure we understand what a spherical triangle looks like and how arc distances on it are measured. Click on the triangle icon to see the video clip.
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Sorry about that interruption. The helmsman needed a little help. The winds had increased in strength and the sails needed to be adjusted. Let's take advantage of that interruption to make sure some key concepts are clear before we continue. Do you understand the triangle that I've drawn on the orange and how it relates to the boat's position on the Earth and Hawaii's position on the Earth? The boat's position, Hawaii's position, and the North Pole form a triangle on the Earth's surface, just like the triangle that I've drawn on the orange's surface. Because that triangle isn't your normal "flat" triangle, the math that you learned back in your Geometry class needs to be upgraded a bit. Distances on a sphere don't have to be measured in units like miles or kilometers. They can be, but they don't have to be, and for what we need to do the math is easier is you don't measure them in miles or kilometers. Because distances on a sphere are actually arc segments on a great circle, they can be measured in degrees. Consider the orange that I cut. Notice that I cut it so that the knife went through the position of the boat, the position of Hawaii, and I also made sure that the cut went through the center of the orange (which represents the center of the Earth). Because I cut it through the center of the Earth, the circle that resulted is called a "great circle". The distance that we're looking for... from our boat to Hawaii... is an arc segment on that great circle. That distance is easiest to represent as the measure of angle that is formed by the three points shown in the picture on the right, below: (1) the boat's location, (2) the center of the Earth, and (3) Hawaii's location. If we can find that angle then we can easily calculate how many miles the arc distance is. There's a simple conversion factor that I will tell you a little later. So... do you understand now that the length of an arc on the Earth's surface is easiest to work with if it's measured in degrees, not miles? And do you understand that the measure of an arc's distance is equal to the angle subtended by that arc?.. the angle having its vertex at the center of the Earth and its two sides passing through the two locations we're interested in? If so, then we can continue. If not, then re-study the photographs shown above and re-read the text shown up above before moving on. This concept is something that most people have to think about several times before it really sinks in and it's critically important that you understand it or else you won't get anything that I'll say from this point forward. Click on the triangle icon to see the video clip.
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Question 3: What is the length of arc "c", measured in degrees? Need help? Remember that the modified Law of Cosines that we need to use for spherical triangles is cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C). Solve this for little c to get... Don't forget that when using this formula, all values are measured in degrees. Even the arc distances. OK... so now we know the distance from the boat to Hawaii measured in degrees. We need it in nautical miles. How do we convert it to nautical miles? On the planet Earth, the arc distance in nautical miles is equal to 60 times the arc distance's measure in degrees. (One nautical mile is defined as one minute of arc, so 60 nautical miles would equal one degree of arc.) Actually... that's not quite correct. It used to be correct... or at least people thought so back hundreds of years ago when the nautical mile was first defined. However in recent years scientists have discovered that the Earth isn't a perfect sphere and so some minor adjustments had to be made to the definition of the nautical mile. The new definition of the nautical mile is that it equals 1.0007 minutes of arc, but hey... let's not quibble about an error that's less than one tenth of one percent. For what we're doing here, let's just say that one minute of arc equals one nautical mile and not worry about the small error. Question 4: What is the length of arc "c", measured in nautical miles? That is the distance we need to go to get to Hawaii.Need help? Take the arc distance in degrees that you calculated above and multiply it by 60. (Or if you want to be really accurate, multiply it by 60,04) Now... What about the direction we need to steer the boat. How do we calculate that?To calculate the direction, just think back to how you calculated angle "B" way back at the top of this web page. What you did up there was to rewrite the Law of Cosines in terms of side "b" and angle "B". Then you used your algebra skills to rearrange that Law of Cosines to solve for angle "B". We're going to do exactly the same thing now, just with the spherical version of the Law of Cosines. When we re-write the spherical version of the Law of Cosines in terms of arc "b" and angle "B", we get... cos(b) = cos(a)cos(c) + sin(a)sin(c)cos(B). With a little bit of algebra skill we can solve that for B, to get... Don't forget that when using this formula, all values are measured in degrees. Even the arc distances Question 5: What is the measure of angle B for us, as we bob around in the ocean wondering what direction we need to go to get to Hawaii? Need help? Don't forget that when using this formula, all values are measured in degrees.. even the sides of the triangle. The lengths of those triangle sides are arc distances, not linear distances. So don't go plugging in the number of nautical miles when you enter the value for arc "b". Use the angular measure that you got as an answer to Question 3. Question 6: What should our numeric heading be? Need help? Don't make the mistake of thinking that angle B is equal to our numeric heading. Remember that headings are equal to the number of degrees starting from north and counting clockwise. (Remember that south = 180 ^{o} and west = 270^{o}, so our heading needs to be somewhere in between those two numbers.)OK, now prove to me that you really understood all of that.In order to earn your extra credit on this set of problems, I want you to prove that you actually understood all of that by solving a different problem... one where the boat is in a different location. Question 7: If a boat is located at 42 ^{o} north latitude and 129^{o} west longitude, what distance does it need to go to get to Hawaii and what numeric heading does it need to steer? (Don't try to correct for magnetic variation. Calculate the "true" heading.)Need help? Remember that Hawaii's coordinates... or more precisely, the coordinates to Kaneohe bay on Oahu, which is the destination of this particular sailing trip... are 21 ^{o} north latitude and 159^{o} west longitude.Congratulations! If you made it all the way through this and actually understood it, you've accomplished something that few people ever do... be they landlubbers or sailors. |