Machinery's Handbook, 31st Edition
100 SOLUTION OF OBLIQUE TRIANGLES Solution and Examples of Oblique Triangles (US Customary or Metric Units) One Side and Two Excluded (Not Between) Angles Known (Law of Sines):
If side a , angle A opposite it, and angle B , are known: C 180 ° A B + ( ) – = b a B sin A sin = ------- c a C sin A sin = -------- Area ab C sin 2 = --------- If angles B and C are known, but not A , then A = 180 - ( B + C ). Example: a = 5 cm; A = 80 ° ; B = 62 ° C 180 ° 80 ° 62 ° + ( ) – 180 ° 142 ° – 38 ° = = = b a B sin A sin ------- 5 sin 62 ° × sin80 ° ------------- 5 0.88295 × 0.98481 -------------- = = = = 4.483 cm c a C sin A sin -------- 5 sin 38 ° × sin80 ° ------------- 5 0.61566 × 0.98481 -------------- = = = = 3.126 cm If sides a and b , and angle C between them are known: A tan a C sin b a C cos ( ) – = --------------- , so A = tan –1 a C sin b a C cos ( ) – --------------- B 180 ° A C + ( ) – = c a C sin A sin = -------- Side c may also be found directly as below: c a 2 b 2 2 ab C cos ( ) = + – Area ab C sin 2 = --------- Example: a = 9 inches; b = 8 inches; C = 35 ° A tan a C sin b a C cos ( ) – --------------- 9 sin35 ° × 8 9 cos35 ° ( × ) – --------------------- =
A
b
c
(Known) B (Known)
C
a
(Known)
One Side and Two Excluded Angles Known
b
c
C a 5
B 62°
Side and Angles Known
Two Sides and Included Angle Known:
A
b
c
C B
(Known)
a
(Known)
Two Sides and One Included Angle Known
= 9 0.57358 × 8 9 0.81915 ( × ) – = --------------------- 5.16222 0.62765
B
c
= ---------- 8.22468 =
A
C
b 8 35°
Hence,
A = B 180 ° A C + ( ) – 180 ° 118 ° 4 ′ – 61 ° 56 ′ = = = c a C sin A sin -------- 9 0.57358 × 0.99269 -------------- 5.2 inches = = = tan –1 (8.22468) = 83 ° 4 ′
Sides and Angle Known
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online