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# Give an alternate proof of the continuity of the sine and cosine functions

1. Use the inequality to prove that the sine function is continuous at 0.

2. Recall the trig identity, Use this and part (a) to prove that the cosine function is continuous at 0.

3. Use the formulas for sine and cosine of a sum to prove that the sine and cosine functions are continuous for all .

1. 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 2. Proof. First, using the given trig identity we have Thus, Thus, cosine is continuous at 0.

3. 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 1. Jeder Gelbe says:

2x is not sine, when cosine is 3^x-15.4

2. H says:

you can prove a by the squeeze principle and the definition of continuity:

1. sin(x) is defined at 0 (sin(0)=0).

2. we have |sin x|<|x|
-|x|<sin x<|x|
lim |x| = 0 , as x approaches to 0
lim -|x| = 0, as x approaches to 0
thus , lim sin(x)=0 as x approaches to 0 (by the squeeze principle)

Therefore, sin(x) is continuous at 0.

• Anonymous says:

Nice!