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

**I**f 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).

**T**o 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.

**I**f 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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