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Angle Side Side

Recently, I was at the 2012 AMATYC conference and heard many great ideas for my classroom.  One idea was this,

Don’t use Law of Sines for Angle Side Side, just use Law of Cosines and then apply the quadratic formula.

What a revelation that was!  My students never seem to get the ambiguous case.  Maybe it is the way I try to explain it. Overall, it seems like a bunch of hoops to them.  But by using the quadratic formula students refresh their memory of the quadratic formula and also apply some basic trig.  So, let’s look at an example.

Let the angle be $20.5^{\circ}$ the adjacent side be 31 and the opposite side be 12. Going straight to the Law of Cosines we get the following:

$12^2 = b^2+31^2-2*12\cos(20.5^{\circ})b$ now set the equation equal to zero and we get
$0=b^2-2*12\cos(20.5^{\circ})b + (31^2-12^2)$ now find the values for the quad formula
$a=1, b=2*12\cos(20.5^{\circ}), c= 817$.  Since the discriminant is greater than zero there will be two solutions or in other words, two lengths for the third missing side. Once you have the two sides, then use the Law of Sines to find the angles. Voila, you are done!

I have created a Desmos graph that can be used to further model the idea of Angle Side Side. (See below) In the graph, you can move the opposite side, which is the ‘c slider”, to try and connect with the missing side.  Through careful observation you will notice that the value of the ‘c slider’ that makes a triangle actually is the +/- discriminant/(2a) part of the quadratic formula!  Having students discover this correlates to the idea how the quad formula is the vertex +/- distance to x-intercepts. In this case, the quadratic formula is the distance to the height +/- distance to opposite side. So when the discriminant is imaginary this means there is not a distance to the opposite side. If the discriminant is zero, then the opposite side is actually the height, altitude, of the triangle. Finally, when the discriminant is greater than zero there are two distances to the opposite. I think this is a great bridge back to x-intercepts and reinforces the importance of the quadratic formula.

Check out the desmos graph by clicking on the below picture. It is the Angle Side Side tutorial I mentioned above. Let me know what you think. Will you still use the Law of Sines?