SOLUTION OF OBLIQUE TRIANGLES Machinery's Handbook, 31st Edition
101
Two Sides and the Angle Opposite One of the Sides Known:
If angle A , opposite side a , and other side b are known: B sin b A sin a = ------- C 180 ° A B + ( ) – = c a C sin A sin = ------- Area ab C sin 2 = --------- If B > A but < 90 ° , a second solution, B 2 , C 2 , c 2 , exists: B 2 = 180° - B , C 2 = 180° - ( A + B 2 ) c 2 = ( a sin C 2 ) / sin A, area = ( ab sin C 2 )/2 If a ≥ b sin A , then only the first solution exists. If a < b sin A , then no solution exists. Example: a = 20 cm; b = 17 cm; A = 61 ° B sin b A sin a -------- 17 sin 61 ° × 20 --------------- = = 17 0.87462 × 20 = ---------------- 0.74343 = Hence, B = ° 1 ′ C 180 ° A B + ( ) – 180 ° 109 ° 1 ′ – 70 ° 59 ′ = = = c a C sin A sin ------- 20 70 ° 59 ′ sin × 61 ° sin ------------------- 20 0.94542 × 0.87462 ---------------- = = = = 21.62 cm sin –1 (0.74343) = 48 If all three sides a , b , and c are known, then any angle can be found: A cos b 2 c 2 a 2 + – 2 bc = -------------- B sin b A sin a = -------- C 180 ° A B + ( ) – = Area ab C sin 2 = ----------
A
B C (Known) b
c
a (Known)
Two Sides and Angle Opposite One Side Known
c
B C
a 20
Sides and Angle Known
All Three Sides Known:
b
A
a (Known) C B All Three Sides Known
Example: a = 8 in.; b = 9 in.; c = 10 in. A cos b 2 c 2 a 2 + – 2 bc -------------- 9 2 10 2 8 2 + – 2 9 × 10 × ---------------- = = 81+100– 64 180 = ----------------- 117 180
= ----- = 0.65000
a 8 C B A Sides Known
A = B sin b A sin a =
49 ° 27 ′
Hence,
cos –1 (0.65000) =
-------- 9 0.75984 × 8 =
-------------- 0.85482 =
Hence, = C 180 ° A B + ( ) – 180 ° 108 ° 11 ′ – 71 ° 49 ′ = = = B 58 ° 44 ′ sin –1 (0.85482) =
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