1. Introduction. So we have investigated several useful trig limits, lets consider the bigger problem: differentiating trigonometric functions.
We will use the following properties without proving them:
Where we have be arbitrary real numbers.
2. Derivative of Cosine. We see from (1a) that
We divide both sides by obtaining
Taking the limit as , and recalling our useful trigonometric limits, we have the definition of the derivative for cosine
3. Derivative of Sine. Likewise, using Equation (1b), we find
Dividing through by yields
Again, taking the limit yields the derivative
Using our beloved trig limits produces:
Thus we obtain
Exercise 1. Prove .
Exercise 2. Recall the secant function is . What is its derivative?
Exercise 3. Recall the cosecant function is . What is its derivative?
Exercise 4. What is the derivative of with respect to ?