135. Many interesting identities can be established connecting the functions of the three angles A, B, C, which satisfy the relation A+B+C=180°. In proving these it will be necessary to keep clearly in view the properties of complementary and supplementary angles. [Arts. 39 and 96.] From the given relation, the sum of any two of the angles is the supplement of the third; thus A B C Again, + + = 90°, so that each half angle is the comple 2 2 2 ment of the sum of the other two; thus Example 1. If A+B+C=180°, prove that sin 24+ sin 2B + sin 2C-4 sin A sin B sin C. The first side=2 sin (A + B) cos (A – B) + 2 sin C cos C Example 2. If A+B+C=180°, prove that tan A+tan B+tan C-tan A tan B tan C. whence by multiplying up and rearranging, tan A+tan B+tan C=tan A tan B tan C, 1. sin 24 - sin 2B+ sin 2C-4 cos A sin B cos C. 2. sin 24-sin 2B- sin 2C-4 sin A cos B cos C. 7. tantan+tan tan+tantan-1. 2 9. 10. cot cos 24+ cos 2B+cos 2C+4 cos A cos B cos C+1=0. cot B cot C+ cot Ccot A+cot A cot B=1. 11. (cot B+cot C) (cot C+cot A) (cot A+cot B) 12. cos2 A+ cos2 B+cos2 C+ 2 cos A cos B cos C=1. =cosec A cosec B cosec C. 14. cos2 24+ cos2 2B+cos2 2C=1+2 cos 24 cos 2B cos 20. 136. The following examples further illustrate the formulæ proved in this and the preceding chapter. cos (4+15°) cos (4 – 15°) – sin (A +15°) sin (A – 15°) NOTE. In dealing with expressions which involve numerical angles it is usually advisable to effect some simplification before substituting the known values of the functions of the angles, especially if these contain surds. Prove the following identities: 1. cos (a+B) sin (a-3)+cos (B+y) sin (8-y) +cos (y+d) sin (y − 8)+cos (+a) sin (8 − a)=0. 4. sin a cos (B+y) − sin ß cos (a+y)=cos y sin (a – ß). 5. cos a cos (B+y) — cos ẞ cos (a+y)=sin y sin (a− ß). 6. (cos Asin A) (cos 24 - sin 2A)=cos A – sin 34. prove that a cos 20+b sin 20=a. [See Art. 124.] 14. (2 cos A+1) (2 cos A − 1) (2 cos 24 − 1)=2 cos 44 +1. 15. tan (B-)+tan (y− a)+tan (a–B) =tan (ẞ− y) tan (y — a) tan (a – B). =1+2 cos (B− y) cos (y − a) cos (a — ß). 18. cos2 a+cos2 B-2 cos a cos ẞ cos (a+B)=sin2 (a+ß). 19. sin2 a+sin2 ß+2 sin a sin ẞ cos (a+B)=sin2 (a+B). 20. cos 12° + cos 60°+cos 84° = cos 24° + cos 48°. If A+B+C=180°, shew that 25. tan ẞ tan y+tan y tan a +tan a tan ẞ= 1. |