CHAPTER XVI. ON THE RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE. 231. The three sides and the three angles of any triangle, are called its six parts. By the letters A, B, C we shall indicate geometrically, the three angular points of the triangle ABC; algebraically, the three angles at those angular points respectively. By the letters a, b, c we shall indicate the measures of the sides BC, CA, AB opposite the angles A, B, C respectively. 232. I. We know that, A + B + C = 180o. [Euc. 1. 32.] have (i) sin A must be positive (and less than 1), (ii) cos A may be positive or negative (but must be numerically less than 1), (iii) tan A may have any value whatever, positive or negative. 234. Also, if we are given the value of (i) sin A, there are two angles, each less than 180", which have the given positive value for their sine. (ii) cos A, or (iii) tan A, then there is only one value of A, which value can be found from the Tables. 235. A A B C = A 2 and its Trigonometrical Ratios are all positive. Also, is known, when the value of any one of its Ratios is given. Similar remarks of course apply to the angles B and C. Example 1. To prove sin (A+B)=sin C. A+B+C=180° .. A+B=180° – C, and .. sin (A+B)=sin (180°C)=sin C. [p. 104.] [Art. 118.] sin 4+Bsin (90-)=cos 2 Example 3. To prove sin A+ sin B+sin C-4 cos COS COS [Art. 157.] 2 Find A from each of the six following equations, A being an Prove the following statements, A, B, C being the angles of a Α A COS 2 (20), sin.cos + sin cos+sin. 008 2 COS B B C C COS 2 2 (25) sin 24+ sin 2B+ sin 2C-4 sin A. sin B. sin C. (26) sin A. cos A - sin B. cos B+sin C. cos C =2 cos A. sin B. cos C. = 4 cos A. sin B. cos C. (27) sin (B+C-A)- sin (C+ A- B) + sin (A+B-C) (28) cos 24+ cos 2B+cos 2C-1-4 cos A. cos B. cos C. (29) sin2 A-sin2 B + sin2 C=2 sin A. cos B. sin C. (30) cos (B+C-A)+cos (C+A-B) - cos (A+B-C)+1 =4 sin A. sin B. cos C. COS =sin 4. sin. sin + 1. 2 (32) tan A+tan B+tan C÷tan A. tan B. tan C. |