Constructivism in math

Monica wrote to me recently:

In some of my math rules, it says that cos(-t)=cos(t) and sin(-t)=-sin(t)Why is the cos not changed to -cos(t) like the sin function is changed?

I replied:

Hello Monica

Grab your calculator and try the following.

sin 35° (write down the answer)

sin (-35°) (write down the answer - what can you conclude?)

Now, try it again, this time:

sin 88°

sin (-88°) Did the same thing happen?

Then try these 2, with cosine:

cos 50° (write the answer)

cos (-50°) (what can you conclude this time?)

To check your conclusion consider this pair:

cos 20°

cos (-20°)

These sort of "rules" are given to you in your text book as a summary of what someone discovered in the past. This is a good example where it is better that you discover it yourself - then the rule will make sense and you are more likely to remember it too.

Of course, we cannot get students to discover everything they need to know (you would never get through all the content).

But when math lessons are 100% rules- and formula-based, it is not surprising to get a question like this.

See the 3 Comments below.