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Trigonometry The word trigonometry literally means “measure of triangles.” Its fundamental objects are the trigonometric ratios for right triangles: If ΔABC has right angle at C, and as is customary, we label the angles of the triangle by the names of their vertices and the sides by a = BC,b = CA, c = AB, € then a sin A = € is the sine, c b cos A = is the cosine, and c a tan A = is the tangent b of (acute) angle A. We can also define cscA (the cosecant), secA (the secant), and cotA (the € cotangent) of A to be the corresponding reciprocals of the ratios for sine, cosine and tangent. To extend these definitions to angles other than acute angles, define sin A = sin(180 – A) and cos A = –cos(180 – A) for any obtuse angle A (definitions for the other trig functions can be deduced as ratios of these). Further, to make the sine and cosine continuous functions, we put sin 0 = 0, sin 90 = 1, sin 180 = 0; and cos 0 = 1, cos 90 = 0, cos 180 = –1. Many relationships in trigonometry hinge on the Theorem [The Pythagorean Theorem] If ΔABC has right angle at C, then a 2 + b 2 = c 2. // Corollary [The Pythagorean Identity] If ΔABC € has right angle at€C, then sin 2 A + cos 2 A = 1. // In fact, the Pythagorean Identity can € be used to give a proof of the € Theorem [The Law of Sines] If ΔABC is any triangle, then a b€ c = = . // sin A sin B sinC Theorem [The Law of Cosines] If ΔABC is any 2 2 2 triangle, then a + b − 2ab cosC = c . // € Corollary [The Cevian Formula] In ΔABC , if € D ∈ A€B , and the segment C D , called a cevian of the triangle, has length d = CD, which cuts A B into parts with ratios p = AD/AB and q = DB/AB, € 2 2 then pa 2 + qb 2 = pqc + d . // € € € € Corollary In ΔABC , if M is the midpoint of A B , then the length of the median C M is € CM = 1 2 a 2 € € + 12 b 2 − 14 c 2 .€//