How Far Can Perfect Eyes See? An Ideal Earthly Scenario

Logic, Mathematics
[caption id="attachment_26540" align="alignright" width="480"] Image by Meg Learner[/caption]I once met a fellow in Virginia who said, 'The human eye is an amazing thing. Why, if there were no mountains in-between, we could see California!' Of course, he was referring to perfect eyes... Of course, that is just plain nuts. Nevertheless, it raised the question, "Just how far away could a person see an object if nothing interfered? Let's consider the answer to that question. Conditions First, we need to set conditions or ground rules. Earth is sufficiently round to call it a sphere, so we treat it as such. In fact, we assume it is perfectly smooth even to an ant. Further, we assume the atmosphere is perfectly clear, and the observer has perfect eyes. When we look out toward…
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Deriving Basic (Circular) Trigonometric Functions

Mathematics, Physics
[caption id="attachment_18674" align="alignright" width="401"] Figure 1[/caption] Draw two intersecting lines in space, illustrated below. The mathematician will not want to leave this simple drawing without completely pointing out its features and labeling those features. We do so to begin our understanding of basic (circular) trigonometric functions. [caption id="attachment_18675" align="alignleft" width="387"] Figure 2[/caption] We label the point of intersection of course – P will do. But the intersection produces what looks like slices in a pie. The size of those slices of pie were determined by how the two lines intersected, how "wide apart" the lines are. We label these as angles α (alpha) and β (beta). Superimposing a Circle [caption id="attachment_18676" align="alignright" width="387"] Figure 3[/caption] The title of this paper is understanding basic circular trigonometric functions. So at this stage,…
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What are Radians? Where Do They Come From?

Consider a simple equilateral triangle (a triangle that has 3 equal sides and 3 equal angles). Most high-school students know the three angles of such a triangle are 60 degrees (60°) each, for a total of 180°. But degrees is not the only unit used to quantify an angle. Alternately radians can be used. What are radians? Are they just another number? Where do they come from? Degrees Before we get into radians, however, let’s consider where degrees came from, and why it may not be the best choice for the measurement of an angle. If you are facing north and turn to the east, you have turned 90 degrees. Now turn south and you’ve turned another 90 degrees. Turn west, another 90 degrees. Continue the turn so you once…
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