- Use the inequality
to prove that the sine function is continuous at 0.
- Recall the trig identity,
Use this and part (a) to prove that the cosine function is continuous at 0.
- Use the formulas for sine and cosine of a sum to prove that the sine and cosine functions are continuous for all .
- Proof. To show is continuous at 0, we must show that . We show this limit is zero directly from the epsilon-delta definition of the limit, i.e., given arbitrary positive , we have whenever . Let be an arbitrary number greater than 0, . Then, let . Using the given inequality we have,
Thus, is continuous at
- Proof. First, using the given trig identity we have
Thus, cosine is continuous at 0.
- Proof. Finally, to show sine and cosine are continuous for all , we show that , and . First, we recall the formulas for the sine and cosine of a sum,
So, we compute the limits
Therefore, sine and cosine are continuous for all