{"id":5664,"date":"2013-03-28T11:36:26","date_gmt":"2013-03-28T11:36:26","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=5664"},"modified":"2014-12-02T08:27:04","modified_gmt":"2014-12-02T08:27:04","slug":"sines-cosines-and-tangents","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/sines-cosines-and-tangents\/","title":{"rendered":"Sines, Cosines, and Tangents"},"content":{"rendered":"

Overview:\u00a0 Introduction to Trigonometric Functions<\/strong>
\nAn angle is formed when a ray is rotated in a coordinate plane around the x axis (the horizontal axis).\u00a0 The measurements of the length of the sides of a triangle have a special relationship to one another, called the sine, the cosine, and the tangent.\u00a0 They are ratios, similar to the slope and y intercept.<\/p>\n

Review of Right Triangles<\/strong>
\nWhen a ray is rotated in the coordinate plane around the x axis, one can always make the angle into a triangle by dropping a perpendicular line from a point on the ray to the x axis.\u00a0 The perpendicular line will form a 900<\/sup> angle for one of the angles of the triangle.\u00a0 By the Euclidean definition, the sum of all three angles will equal 180o<\/sup>.\u00a0 Therefore, the other two angles will equal 90o<\/sup>.\u00a0 In addition, the sides of the right triangle are in a special relationship, the Pythagorean theorem, a2<\/sup> +b2<\/sup> = c2<\/sup>.\u00a0 By definition, a2 <\/sup>is the opposite side to the right angle, b2<\/sup> is the adjacent side, and c2<\/sup> is the hypotenuse.<\/p>\n

Quadrants and Coordinates<\/strong>
\nThe triangle will be in different quadrants in the coordinate plane depending on where the original ray is located.\u00a0 Since the triangle starts at the origin of the x axis and the y axis, hypothetically speaking, if all points along the ray are positive, the triangle is in Quadrant 1.\u00a0 The angle formed by the opposite side to the right angle will be from 0o <\/sup>to 90o<\/sup>.\u00a0\u00a0\u00a0 If the x coordinate is negative but the y coordinate is positive, the triangle will be in Quadrant 2, and the angle formed by the opposite side to the right angle will be from 90o<\/sup> to 180o<\/sup>.\u00a0 If both the x and y coordinates are negative, the triangle formed will be in Quadrant 3, and the angle formed by the opposite side to the right angle will be from 180o <\/sup>to nearly 270o<\/sup>.\u00a0 When the x coordinate is positive but the y coordinate is negative, the triangle will be in Quadrant 4, and the angle formed by the opposite side to the right angle will be from 270o<\/sup> to nearly 360o<\/sup>.<\/p>\n

Relationship Between Sines, Cosines, and Tangents<\/strong>
\nThe sine is the ratio of the opposite side to the hypotenuse. If the angle measures from 0o<\/sup> to 90o<\/sup>, that ratio will increase from 0 to nearly 1.\u00a0 At 90o<\/sup>, the sine equals 1.\u00a0 From just over 90o<\/sup> to 180o, <\/sup>it decreases from 1 to nearly 0.\u00a0 From 180o<\/sup> to nearly 270o<\/sup>, the sine decreases further \u00a0from 0 to -1.\u00a0 At 270o<\/sup>, the sine measures -1, and from 270o<\/sup> to 360o<\/sup>, the sine increases from -1 to 0.<\/p>\n

The cosine is the ratio of the adjacent side (the b side) to the hypotenuse.\u00a0 If that angle measures from 0o<\/sup> – 90o<\/sup>, the cosine decreases from 1 to nearly 0.\u00a0 At 90o<\/sup>, the cosine will equal 0.\u00a0 If the adjacent angle measures between 90o<\/sup> and 180o<\/sup>, the cosine will decrease from 0 to -1. From 180o<\/sup> to 270o<\/sup>, the cosine increases from -1 to 0, and from 270o<\/sup> to 360o<\/sup>, the cosine increases from 0 to 1.<\/p>\n

The tangent is the ratio of the opposite side of the triangle to the adjacent side of the triangle.\u00a0 By definition, sines and cosines are real numbers, between -1 and 1.\u00a0 Tangents are undefined for degrees 90o<\/sup> and 2700<\/sup>, but they increase from 0 to very large numbers in all other quadrants.<\/p>\n

Applications of Sines, Cosines, and Tangents<\/strong>
\nTrigonometric functions have many applications in the real world, because they are the result of mathematical laws.\u00a0 Harmonics such as sound waves or light waves follow sines and cosines.\u00a0 Triangulation and navigation apply trigonometry to determine unknown distances.\u00a0 Fields as diverse as building bridges and astronomy depend upon trigonometry.<\/p>\n

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SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Halifax, NS, Canada visit: Tutoring in Halifax, NS, Canada<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Overview:\u00a0 Introduction to Trigonometric Functions An angle is formed when a ray is rotated in a coordinate plane around the x axis (the horizontal axis).\u00a0 The measurements of the length of the sides of a triangle have a special relationship to one another, called the sine, the cosine, and the tangent.\u00a0 They are ratios, similar […]<\/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":[18],"tags":[406,415,1649,1806],"class_list":["post-5664","post","type-post","status-publish","format-standard","hentry","category-trigonometry","tag-coordinate-plane","tag-cosine","tag-sine","tag-tangent"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5664","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=5664"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5664\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=5664"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=5664"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=5664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}