As a high school math teacher, I find it really important to help students link the new concepts they are learning to the knowledge they already have. This helps students get a fuller picture of math and that math isn’t just a bunch of formulas they need to remember.
One example of this is the cosine law and the pythagorean theorem.
Imagine a triangle with sides A,B, and C, with angles \alpha, \beta, and \gamma. It may look like the one pictured below.
The cosine law states the following: c^2 = a^2 + b^2 -2abcos\gamma
The cosine law looks very similar to another formula that the students most likely have seen before which is the Pythagorean Theorem.
Imagine a right triangle with sides A,B, and C like the triangle pictured below.
The pythagorean theorem states the following: c^2 = a^2 + b^2
The only difference between the cosine law and the pythagorean theorem is that the cosine law has an additional -2ab\gamma.
Why doesn’t the pythaorean theorem have the -2ab\gamma? Well that’s because in the right triangle, \gamma=90^o and cos 90^o=0. So -2abcos\gamma = 0 for a right triangle.
In other words, the Pythagorean Theorem is a special case of the cosine law where the triangle is a right triangle.
By showing how the cosine law and pythagorean theorem relate to each other, students understand that it is in fact, one concept and that a lot of ideas in math are relatedd to one another.
Unfortunately, math is commonly taught as a series of formulas and rules. But teaching math is so much more than that. It’s about coming to an understanding of why these formulas must be and what are the implications of these formulas. The more one learns about math, the more interesting math becomes.