Already, we have a bit of a mystery.\u00a0\u00a0Students don’t know where the word “sine” comes from, but that’s fine.\u00a0 Lots of new topics introduce new vocabulary without going into the etymology of every new term.\u00a0 But what about “cosine”?\u00a0 What is co- about it?\u00a0 A minor mystery to be sure, but even so…<\/p>\n Then we\u00a0tell them about “tangent”.\u00a0 We say it\u00a0is\u00a0“opposite over adjacent”, which is the TOA in SOHCAHTOA.\u00a0 And a tiny bit of algebra shows that the tangent\u00a0can also be\u00a0written as sine over cosine.\u00a0 But we still have a mystery:<\/p>\n What is “tangent” about the tangent?\u00a0<\/strong> <\/em><\/p>\n This is not just an etymology question.\u00a0 The word “tangent” is already part of\u00a0our math vocabulary.\u00a0 We saw in geometry that at tangent line is a line that touches but does not cross a circle.\u00a0 There are all kinds of rules about the angles and the lengths\u00a0associated with\u00a0tangent lines (and secant lines too, for that matter).\u00a0 But what does that use of the word have to do with the way we are using it now, here in trigonometry?<\/p>\n From there, we move on to the reciprocal trig ratios:<\/p>\n And now you have more mysteries:<\/p>\n What does this<\/span> secant have to do with the secant we learned about in geometry?<\/strong><\/em><\/p>\n In geometry class, a secant is a line that crosses a circle.\u00a0 Is there some reason we are re-purposing this vocabulary?<\/p>\n And as long as we are asking questions:<\/p>\n Why did you match “secant” with “one over COSINE”? And “cosecant”\u00a0 with “one over SINE”?\u00a0 Isn’t that making things unnecessarily complicated?<\/strong><\/em><\/p>\n And once again:<\/p>\n What is “co-” about the cosecant?\u00a0 And the cotangent?<\/strong><\/em><\/p>\n All of these mysteries can be resolved with a single diagram.\u00a0 I will develop the diagram step-by-step in my next post.\u00a0 But for now, here it is in the finished form.\u00a0 Ponder it well…<\/p>\n As you will see in the next posts, there are lots of similar triangles waiting to be discovered in this diagram.\u00a0 Also, the Pythagorean theorem makes several useful appearances.<\/p>\n But for now, let’s resolve one of the mysteries:<\/p>\n Look at the segments labeled sine, secant and tangent.\u00a0 Note their positions relative to the central angle, \u03b8.<\/p>\n Now look at the segments labeled cosine, cosecant and cotangent.\u00a0 Note their positions relative to the central angle 90-\u03b8.\u00a0 Each of these segments relate to 90-\u03b8 the way their corresponding segments relate to \u03b8. And remember that 90-\u03b8 is called the “complement” of the angle.\u00a0 So that’s where these segments get their names:<\/p>\n “the complement’s sine” —> cosine<\/p>\n “the complement’s secant” —> cosecant<\/p>\n “the complement’s tangent” —> cotangent<\/p>\n One mystery solved: “co-” is for “complement” !<\/strong><\/em><\/p>\n More mysteries to be unraveled soon.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" One of the digressions we follow almost every time I teach AP Physics is intended to give my\u00a0students a better visual understanding of the 6 basic trig ratios.\u00a0 But it\u00a0also resolves\u00a0several mysteries left over from their first introduction to trig. … Continue reading ![\"post25pic1\"](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2014\/12\/post25pic1.png\")
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