dan's math@home - lessons - precalculus
Solving General Triangles:
[ Pythagorean Theorem | Law of Sines | Law of Cosines ]
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(c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.

If you know two sides of a right triangle, you can find the third by using the

Pythagorean Theorem: a^2 + b^2 = c^2. (back to trig page)

The four triangles have been moved around, but the "uncovered" yellow area is the same in both pictures below; a^2 + b^2 on the left and c^2 on the right.

This cool picture got me my job; I used it in my interview "mini-lecture."

Some examples of [a, b, c] "Pythagorean Triples" are:
[3, 4, 5] because 3^2 + 4^2 = 5^2 (check that 9 + 16 = 25.). Others are:
[5, 12, 13] , [6, 8, 10] , [9, 12, 15] , [8, 15, 17] , [7, 24, 25] , and [20, 21, 29].
The list goes on forever, and there are formulas to help generate Pythriples too:
If m > n are natural nos, then if a = m^2 - n^2 , b = 2mn , c = m^2 + n^2 ,
then [a, b, c] is a Pythag Triple (a^2 + b^2 = c^2 is guaranteed by algebra).

To solve for unknown triangle parts, you can often use the Law of Sines:

In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :

(sin A) / a = (sin B) / b = (sin C) / c . (back to trig page)

Proof: What can we do if there's no right angle? Make one! If we want to relate the sides a, b, c with the angles A, B, C in the diagram below, we drop a perpendicular from A to a base point D along side a ...

... now we have a right triangle ADB , so sin B = AD/AB . But AB = c , so we get the vertical dotted line AD = c sin B. Now AD is opposite both angles B and C , so we write it two ways: AD = c sin B = b sin C . By dividing we get

THE LAW OF SINES : a / (sin A) = b / (sin B) = c / (sin C) .

We really only proved the last equality but the other part works exactly the same way.

If you know two sides and the included angle, there's the Law of Cosines:

In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :

c^2 = a^2 + b^2 - 2ab cos C . (back to trig page)

This is cool because if C is a right angle then cos C = 0 and we get the old Pythm.

Here's an animation I did in Mathematica (to QuickTime to GIF Builder) to show two sticks of length 3 and 4; the angle changes from 0 to 180 degrees and the distance between the tips changes as shown. Think of a rubber band changing length as the sticks spread apart!

Remember you can find c by using c^2 = a^2 + b^2 - 2ab cos C.

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