Using the Law of Cosines
Date: 05/21/98 at 05:15:17 From: Dan Subject: Trigonometry In triangle ABC, side a = 30m, side b = 36m, and side c = 10m. The question asks to find the size of the angle between sides b and c. I think the question has something to do with advanced trigonometry, but I am unsure of how to go about it. Your help would be appreciated! Thanks, Dan
Date: 05/21/98 at 07:34:11 From: Doctor Anthony Subject: Re: Trigonometry The convention is to use lower case letters a, b, c, to represent the sides of the triangle, and to use upper case letters A, B, C, to represent the angles, with side a opposite angle A, side b opposite angle B, and side c opposite angle C. There are two principal formulae for solving triangles of any shape (not just right-angled triangles). Which one you use depends on what information you are given. You will find these formulae proved in any textbook on elementary trigonometry. Cosine Formula --------------- a^2 = b^2 + c^2 - 2bc*cos(A) or b^2 = c^2 + a^2 - 2ca*cos(B) or c^2 = a^2 + b^2 - 2ab*cos(C) This is used if you are given three sides or if you are given two sides and the included angle. ("Included" means the angle betweem the two given sides). Sine Formula ------------- a b c -------- = -------- = -------- sin(A) sin(B) sin(C) This is used if you are given two angles and a side, or two sides and a non-included angle. In the problem you stated, we are required to find the angle between sides b and c. This is angle A. Since we are given three sides, we use the cosine formula. a^2 = b^2 + c^2 - 2bc*cos(A) 2bc*cos(A) = b^2 + c^2 - a^2 b^2 + c^2 - a^2 cos(A) = ------------------ 2bc Putting a = 30, b = 36, c = 10 we have: 1296 + 100 - 900 cos(A) = ---------------- 720 496 31 cos(A) = ----- = ----- = 0.68888 720 45 A = 46.458 degrees = 46 deg, 27 min, 28 secs If you require the other two angles, it is quicker now to use the sine formula. Also, remember that the three angles add up to 180 degrees, so if you have two angles, you can find the third by subtraction from 180 degrees. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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