30 8420325 8.420804 8.421283 8421761 8422239 8422717 29 31 8 425096 8425570 8.426044 8426517 8.426990 8.427462 28, 32 8429815 8430285 8430753 8431221 8431689 8432156 27 8. 33 84344848°434948 8435412 8435875 8436338 8436800 26 34 8*439103 8.4 439562 844002 I 8*440479 8.440937 8.4 '441394 25 35 8443674 8444128 8444582 8445035 8445488 8.445941 24 36 84481968448646 8449095 8449544 8449992 8°450440 23 37 84526738453118 8453562 8454006 8454450 8454893 22 38 8457103 8457543 8457984 8458423 8458863 8459301 21 39 8461489 8461925 8-462360 8462796 8.463230 8463665 20 40 8465830 8466262 8466693 8 467124 8467555 8.467985 19 41 8470129 8.470557 8°470984 8471410 8471837 8.472263 18 428474386 8474809 8475232 8475655 8.476077 8476498 17 43 8478601 8479020 8479439 8479857 8480276 8.480693 16 44 8482776 8483191 8483606 8484020 8.484434 8.484848 15 45 8486910 8487322 8487733 8488143 8488553 8.488963 14 46 8491006 8491414 8491821 8492228 8492634 8493040 13 47 8495064 8495467 8495871 8:496274 8°496676 8.497078 12 488499084 84994848°499883 8500283 8.500681 8.501080 II 49 8.503067 8.503463 8.503859 8504255 8.504650 8.505045 ΙΟ 50 8507014 8507406 8507799 8.508191 8.508582 8.508974 9 518510925 8511314 8511703 8512091 8512480 8.512867 52 8.514801 8.515187 8.515572 8.515957 8.516342 8.516726 53 8.518643 8.519025 8.519407 8.519789 8.520170 8.520551 54 8522451 8.522830 8.523209 8 523587 8.523965 8.524343 55 8.526226 8.526602 8.526977 8°527353 8 527727 8.528102 56 8.529969 8 530341 8530714 8.531085 8.531457 8.531828 3 57 8.533679 8534049 8534418 8534786 8.535155 8.535523 58 8537358 8537725 8.538091 8538456 8.538821 8.539186 59 8.541007 8.541370 8°541733 8°542095 8°542457 8.542819 6" 978 543 2 I 30' 24" 18" 12" Cosine. 88 degrees. [In the following formulæ, a, b, c denote the sides of the triangle, and A, B, C the angles respectively opposite.] I. RIGHT-ANGLED PLANE TRIANGLES. Α B C a In the triangle AB C, let C be the right angle and c the Case 2. Data: The hypothenuse and a side, Given c, a; Sought b, A, B. с sin A=- ; b=c sin B = c cos A. Case 3. Data: A side and one of the acute angles. b= a cot A; c= a sin A ; B = 90° A. Case 4. Data: The hypothenuse and an angle, a = c sin A; b = c cos A; B = 90° — A. M = tan & A = ; tan | B=;tan CM Case 2. Data: Two sides and the included angle. α b tan (A-B) == tan (A + B). a+b Case 3. Data: Two sides and the angle opposite one of Note. When the side b adjacent to the given angle A is greater than a the side opposite to it, there are two triangles which furnish the same data a, b, A; and, consequently, the solution is ambiguous. Case 4. Data: Two angles and the side between them. Given A, B, c; Sought a, b, C. In the triangle A B C, let C be the right angle and c the hypothenuse. Case 1. Data: The two sides. Given a, b; Sought c, A, B. cos c = cos a cos b; tan A= ; tan B = tan a Case 2. Data: The hypothenuse and a side. b; Sought a, A, B. COS C cos α = ; cos A = sin B = Case 3. Data: A side and the angle opposite. Given a, A; Sought b, c, B. Note. This case is ambiguous, each of the things sought having two values, viz. either the angle found or its supplement. sin b = sin (180° — b). Thus, Case 4 Data: A side and the adjacent angle. Given a, B; Sought b, c, A. tan b = sin a tan B; cot c=cot a cos B; cos A = cos a sin B. Case 5. Data: The hypothenuse and an angle. sin a sin c sin A; tan b= tan c cos A; cot B = cos c tan A, Case 6. Data: The two oblique angles. Given A, B; Sought a, b, c. Case 1. Data: The three sides. Given a, b, c; Sought A, B, C. Let s = (a+b+c); (s b) (sin (s — a) sin (e — ¿) sin (e — 0)); sin s |