Finding Angles without Using Trigonometry
Date: 08/27/2001 at 22:01:03 From: Marc Foster Subject: Re: Given 1 side of an isosceles triangle ABC, solve for angle A I know I learned this at one point in time or another, but I can't seem to figure it out. A /\ / \ / \ / \ / \ B/__________\C Where BC = 150 The height of the triangle = ((BC/2)^2)-40 AB Congruent to AC BC does not equal AB Solve for angle A Is this possible? Maybe I should just cut to the chase and ask the real question... Given the lengths of three sides of a triangle, can you determine the measures of the three angles without using cos or sine or even looking at a trig table? In other words, can one solve the angles using only geometry and algebra? Even if the difficulty outweighs the practicality of the method, I would enjoy the peace of mind gained from knowing it. Thanks, Marc Foster
Date: 08/28/2001 at 13:05:04 From: Doctor Rob Subject: Re: Given 1 side of an isosceles triangle ABC, solve for angle A Thanks for writing, Marc. In answer to your "cut-to-the-chase" question, the answer is a qualified yes. If one knows the three sides a, b, and c, and wants angle A, one can compute the quantity x = (b^2+c^2-a^2)/(2*b*c). Then one can find A = Pi/2 - x - (1/2)*x^3/3 - (1/2)*(3/4)*x^5/5 - (1/2)*(3/4)*(5/6)*x^7/7 - ... . Take as many terms as needed to get the accuracy you want. This will give you the angle measure in radians. To convert to degrees, multiply by 180/Pi. (Of course trigonometry is present here, just disguised. x = cos(A), and A = Arccos(x) = Pi/2 - Arcsin(x).) Example: Suppose the sides of a triangle are a = 7, b = 5, and c = 4. Then x = (25+16-49)/(2*4*5) = -1/5, A = Pi/2 + 1/5 + 1/750 + 3/125000 + 1/1750000 + 7/450000000 + ..., = 1.5707963267 + 0.2000000000 + 0.0013333333 + 0.0000240000 + 0.0000005714 + 0.0000000156 + ..., = 1.77215425 radians, rounded to 8 decimal places. In degrees, that is A = 1.77215425*180/3.1415926536 = 101.53696 degrees. Similarly, B = 44.41531 degrees (x = 5/7), C = 34.04773 degrees (x = 29/35). Aside from this kind of approach, you cannot find the angles without using trigonometry. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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