Use an appropriate Taylor polynomial about 0 and the Lagrange Remainder Formula to approximate sin(4/7) with an error less than 0.0001.

sin(4/7) ≈ ? (give your answer to 6 decimal places)

What is the smallest value of n for which the approximation above is guaranteed to have an error less than 0.0001? (Be careful. Think about the actual terms used in the series as well as the remainder.)

n = ?

Question

Use an appropriate Taylor polynomial about 0 and the Lagrange Remainder Formula to approximate sin(4/7) with an error less than 0.0001. 

sin(4/7) ≈ ?  (give your answer to 6 decimal places)

What is the smallest value of n for which the approximation above is guaranteed to have an error less than 0.0001? (Be careful. Think about the actual terms used in the series as well as the remainder.)

n =  ?

anonymous · Answer

Recall that the sin (x) = x - (x^3 / 3!) + (x^5 / 5!) - . . . 

Now plug in (4/7) for x and start computing the first couple terms until you are within .0001 of the actual value of sin(4/7).

anonymous · Answer

is there anyway to do this without a calculator? I'm trying to determine if this type of problem is worth spending time on when we are not allowed to use calculators on our exam. Thanks, @RBauer4

anonymous · Answer

Hmmmm. I'm not convinced that there is a convenient way to go about this problem. It has been awhile since I had Calc II, so I am not certain. However, I'm pretty sure that the sin(4/7) is irrational; so, without approximating it, I'm not sure that you would be able to calculate it's value in order to guarantee your error. Carrying out the first few terms wouldn't be difficult by hand, but it could be time consuming. If you carry out enough terms, then you could observe that the sum is only fluctuating by values more miniscule than the error bound you were given. Again, time consuming and tedious.