The law of cosines in demonstrated in many trigonometry classes by the classic geometric proof shown below.
Consider the triangle ABC.
Construct an altitude, h, of triangle ABC from vertex C to the extension of side AB.
Now consider right triangle BDC. We can see a relationship between angle B, side h, and side a.
Now consider right triangle ADC. We can see a relationship between angle DAC, side h, and side b.
Since angle DAC is the reference angle for angle BAC, sin(DAC) = sin(BAC) . Hence,
Now we have two equations in h so we can eliminate h.
Hence,
Dividing by ab yields
Finally, since angle BAC is angle A in triangle BAC we can rewrite the above as
Generally students are not convinced by any of these equalities or if they are, they may believe that this law works only for the given triangle. In order to convince students of the generality of the law, the instructor can use GSP and the following investigations.
1. Using GSP* construct a right triangle ABC so that angle C is the right angle and side CB is the base of the triangle.
* IF YOU DON'T HAVE GSP, CLICK HERE TO DO THE ACTIVITY.
2. Measure:
3. Calculate the sine of angle B and the ratio of side b to side c.
4. Move point A. What happens to angle B? What happens to sin(B) and b/c?
5. So what is the general relationship between angle B, side b, and side c?
Just in case you can't access GSP through the link above, this is what it looks like.
1. Using GSP* Construct an obtuse triangle in which angle A is the obtuse angle and side AB is the base of the triangle.
* IF YOU DON'T HAVE GSP, CLICK HERE TO DO THE ACTIVITY.
2. Label side BC as side a, side AC as side b and side AB as side c.
3. Measure the sides and the angles of the triangle.
4. Compute the sine of each angle using GSP. Can the sine of the angles be calculated since they are not in a right triangle? How?
5. Measure the sides of the triangle using GSP.
6. How many ratios of sides and the sines of the angles of triangle ABC are possible?
7. Compute the ratios you described in #6 and move point C to find any connections that exist between the ratios.
1. Using GSP* Construct an obtuse triangle in which angle A is the obtuse angle and side AB is the base of the triangle.
* IF YOU DON'T HAVE GSP, CLICK HERE TO DO THE ACTIVITY.
2. Label side BC as side a and side AC as side b.
3. Construct the altitude from angle C to the extension of side AB. Call the point of intersection D.
4. Label side DC as side h.
5. Is there any connection between the angles in triangle DAC and those in BAC?
6. Measure the sides and angles of triangle BDC and DAC and see if you can find any connections? Move point C around and see if you can see any connections. Don't forget to find the sine of some of the angles.
Just in case you can't access GSP through the link above, this is what it looks like.
