{"id":412,"date":"2014-12-18T14:39:03","date_gmt":"2014-12-18T18:39:03","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=412"},"modified":"2014-12-19T08:39:24","modified_gmt":"2014-12-19T12:39:24","slug":"picturing-the-trig-ratios-step-by-step","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=412","title":{"rendered":"Picturing the Trigonometric Ratios: Step-by-step"},"content":{"rendered":"

\u00a0<\/strong><\/p>\n

Let’s start with the unit circle and a\u00a0central angle \u03b8\u00a0that meets an arc on the circle:<\/p>\n

\"post27pic1\"<\/a><\/p>\n

The blue segment is called the “sine” of the angle.\u00a0 It gets its name from a word that means “bowstring”.\u00a0 Here is a diagram of a bow string truss.<\/p>\n

\"post27pic2\"<\/a><\/p>\n

So if you think of the intercepted arc as the bow and focus only on the upper half…<\/p>\n

\"post27pic3\"<\/a><\/p>\n

… you can see how they first decided to call that segment the “bowstring.”<\/p>\n

But you do not have to hold your bow vertically.\u00a0 If you use the complementary angle, 90-\u03b8, you get the horizontal segment I have drawn in red (and re-drawn on the x-axis as well…you’ll see why soon.)<\/p>\n

\"post27pic4\"<\/a><\/p>\n

We can call that red segment the “bowstring of the complement”, or the “complement’s sine.”\u00a0 Or we could just call it the cosine<\/strong><\/em>!<\/p>\n

Now we are going to add two segments to the diagram: one\u00a0that just\u00a0touches\u00a0the circle and one that crosses the circle.\u00a0 We will use the same names we used for these back in geometry class.\u00a0 As you can see from the labels, the orange one is called the tangent<\/strong><\/em> and the green one is the secant<\/strong><\/em>.<\/p>\n

\"post27pic5\"<\/a><\/p>\n

Now, as we examine this diagram, we can find a number of interesting pairs of similar triangles.\u00a0 To discuss them, it would be helpful if we label some points of interest:<\/p>\n

\"post27pic8\"<\/a><\/p>\n

Now we have arrived at the key step.<\/p>\n

Take a moment to convince yourself that\u00a0\u0394EBC is similar to \u0394BDC.\u00a0<\/strong> <\/em><\/p>\n

Then, since similar triangles have proportional sides, it must also be true that:<\/p>\n

\"post27pic7\"<\/a><\/p>\n

These are NOT arbitrarily chosen definitions.\u00a0 They follow by necessity from the geometric definitions of these terms.<\/p>\n

I will pause here to allow everyone to catch their breath.\u00a0 In the next post, we will finish developing the diagram and add in the Pythagorean identities.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a0 Let’s start with the unit circle and a\u00a0central angle \u03b8\u00a0that meets an arc on the circle: The blue segment is called the “sine” of the angle.\u00a0 It gets its name from a word that means “bowstring”.\u00a0 Here is a … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-6E","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/412"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=412"}],"version-history":[{"count":9,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/412\/revisions"}],"predecessor-version":[{"id":432,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/412\/revisions\/432"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=412"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=412"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}