{"id":5746,"date":"2013-04-19T15:07:19","date_gmt":"2013-04-19T15:07:19","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=5746"},"modified":"2014-12-02T08:27:03","modified_gmt":"2014-12-02T08:27:03","slug":"right-triangles-and-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2013\/04\/19\/right-triangles-and-trigonometric-functions\/","title":{"rendered":"Right Triangles and Trigonometric Functions"},"content":{"rendered":"

Overview:\u00a0 What Are Right Triangles?<\/strong><\/p>\n

Right triangles have one angle with a measure of 90o<\/sup>.\u00a0 As can be shown from the axiom in geometry that the sum of all three angles equals 180o<\/sup>, the sum of the other angles have to equal 90o<\/sup>.\u00a0 \u00a0In special right triangles, such as 45o<\/sup>-45o<\/sup>-90o<\/sup>, or 30o<\/sup>-60o<\/sup>-90o<\/sup>, the lengths of sides are in a special relationship.\u00a0 Therefore, the sines, cosines, and tangents are in a close relationship and are easily found.<\/p>\n

The Pythagorean Theorem and Right Triangles<\/strong><\/p>\n

The Pythagorean Theorem, a2<\/sup> +b2<\/sup> = c2<\/sup>, holds true for right triangles.\u00a0 \u00a0This means that the lengths of the sides opposite and adjacent to the right angle, as well as the hypotenuse, \u00a0can be solved using the relationship between measurements.\u00a0 Suppose in a right triangle one side is equal to 5 cm and the hypotenuse is equal to 13.\u00a0 What is the measurement of the other side?\u00a0 Using the Pythagorean theorem , a2<\/sup> = 132 <\/sup>-52<\/sup>\u00a0 , a2<\/sup>=169-25, or a2 <\/sup>=144.\u00a0 Therefore, a = 12, the square root of 144.<\/p>\n

What Are the Sines, Cosines, and Tangents of that Triangle?<\/strong><\/p>\n

In that example triangle, the sine is the ratio of the opposite side to the hypotenuse, or 5\/13.\u00a0 The cosine is the ratio of the adjacent side to the hypotenuse of 12\/13, and the tangent is the ratio of the opposite side to the adjacent side or 5\/12.\u00a0 These relationships follow the rules for the values of sine, cosine, and tangent, no matter \u00a0in which quadrant the triangle can be graphed.<\/p>\n

What Is the Relationship of Sines, Cosines, and Tangents?<\/strong><\/p>\n

When the triangle is a 45o<\/sup>-45o<\/sup>-90o<\/sup> triangle, the sine and cosine will be equal, because the measurements of the opposite side and the adjacent side of the right angle are the same length.\u00a0 Similarly, the tangent will equal 1, because the sine and cosine are equal.\u00a0 In a 30o<\/sup>-60o<\/sup>-90o<\/sup> triangle,\u00a0 the hypotenuse is twice the length of the opposite side of the right angle.\u00a0 Therefore, the sine — the ratio of the opposite side to the hypotenuse — will equal 1\/2.\u00a0 The cosine — the ratio of the adjacent side to the hypotenuse, will equal \u221a3\/2, and the tangent will equal \u221a3\/3.<\/p>\n

Using a Reference Triangle<\/strong><\/p>\n

Every angle in the coordinate plane can be referenced to a similar right triangle that contains it.\u00a0 What that means for the purposes of using trigonometric functions is that the sign, cosine, and tangent can be measured with an example.\u00a0 Those measurements are contained in \u00a0many different reference tables.<\/p>\n

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Overview:\u00a0 What Are Right Triangles? Right triangles have one angle with a measure of 90o.\u00a0 As can be shown from the axiom in geometry that the sum of all three angles equals 180o, the sum of the other angles have to equal 90o.\u00a0 \u00a0In special right triangles, such as 45o-45o-90o, or 30o-60o-90o, the lengths of […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2,18],"tags":[418,1453,1551,1653,1809],"class_list":["post-5746","post","type-post","status-publish","format-standard","hentry","category-algebra","category-trigonometry","tag-cosines","tag-pythagorean-theorem","tag-right-triangles","tag-sines","tag-tangents"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5746","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=5746"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/5746\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=5746"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=5746"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=5746"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}