Trig2 feb 19 - (a) m 2 a = 1 4 (2 b 2 + 2 c 2-a 2 ), (b) m...

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PEM Level C 19 February 2010 Trigonometric Identities Exercises: 1. Let ABC be an acute-angled triangle. Prove that tan A +tan B +tan C = tan A tan B tan C. 2. Let ABC be a triangle. Prove that sin 2 A 2 + sin 2 B 2 + sin 2 C 2 3 4 . 3. Let ABC be a triangle with inradius r and circumradius R. Prove that (a) cos A + cos B + cos C = 1 + 4 sin A 2 sin B 2 sin C 2 ; (b) cos A + cos B + cos C = 1 + r R ; and (c) cos A + cos B + cos C 3 2 . 4. In 4 ABC , let m a , m b , and m c be the lengths of the medians from A , B , and C , respectively. Let G be the centroid of 4 ABC . Derive the following formulas:
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Unformatted text preview: (a) m 2 a = 1 4 (2 b 2 + 2 c 2-a 2 ), (b) m 2 a + m 2 b + m 2 c = 3 4 ( a 2 + b 2 + c 2 ), and (c) AG 2 + BG 2 + CG 2 = 1 3 ( a 2 + b 2 + c 2 ). 5. (a) Prove that the area of 4 ABC can also be computed as follows: ( ABC ) = 1 2 bc sin A (b) Use (a) to derive another formula for the area of 4 ABC ( ABC ) = abc 4 R , where R is the circumradius of 4 ABC ....
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