Evaluate the following summation (make sure your calculator is in radian mode): ∑ i = 1 4 sin ( π i 8 ) \sum_{i=1}^4\sin\left(\frac{\pi i}{8}\right)

So in this problem, we're asked to evaluate the following sum from one to four of this function right here. Now to solve this problem, what we're going to do is we're going to treat this as our function where I is the input, and we need to go ahead and take this and increment it up by one until we get all the way to four. So starting things off, we're going to have the sine of pi times one. So we'll have pi times one, which is just pi over eight. And then we're going to have this plus, and then we're going to go to the sine of and then in this case, I need to increment up by one. So we'll have the sine of, and then we're gonna have two pi over eight. So I'm literally just taking this I here and replacing it with each number until we get all the way up to four. So I've plus the sine, and then we'll have three pi over eight. And then lastly, we're going to have plus the sine of four pi over eight, because four is the last number that we have in this index. So this is what everything's going to look like. And there is some things I can simplify here. The sine of Pi over eight is going to stay the same. And then we have the sine of two Pi over eight. And this can reduce, because two can go into eight four times. So what that we're gonna have is the sine of pi over four. And then we're going to have this plus the sine, and then in this case, three pi over eight is gonna stay the same. So three pi over eight. Then we're going to have plus, and then four can go into eight two times. So we'll have the sine of pi over two. Now these right here are numbers that you could actually find on a unit circle, the sine of pi over four and the sine of pi over two. Alternatively, though, you could just take this and plug it into your calculator. You just wanna make sure that your calculator's in radian mode when you do this. If you plug this into your calculator, you should get approximately 3.01 radians as the result. So this would be the solution to the problem, and that is how you can solve it. So this is how we can find the summations when dealing with these trigonometric functions. Hope you found this video helpful, and let's move on.

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8. Definite Integrals

Riemann Sums

Calculus

8. Definite Integrals

Riemann Sums

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Multiple Choice

Evaluate the following summation (make sure your calculator is in radian mode):
$\sum_{i = 1}^{4} \sin (\frac{π i}{8})$

1.00

1.38

3.01

0.62

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Verified step by step guidance

Identify the summation expression: $ \sum_{i=1}^4 \sin\left(\frac{\pi i}{8}\right) $. This means you need to evaluate the sine function for each integer value of $ i $ from 1 to 4 and then sum the results.

Set your calculator to radian mode, as the angles in the sine function are given in radians. This is crucial for obtaining the correct values of the sine function.

Calculate $ \sin\left(\frac{\pi \times 1}{8}\right) $. This involves substituting $ i = 1 $ into the expression and evaluating the sine of the resulting angle.

Calculate $ \sin\left(\frac{\pi \times 2}{8}\right) $. Substitute $ i = 2 $ into the expression and evaluate the sine of the resulting angle.

Repeat the process for $ i = 3 $ and $ i = 4 $, calculating $ \sin\left(\frac{\pi \times 3}{8}\right) $ and $ \sin\left(\frac{\pi \times 4}{8}\right) $ respectively. Finally, sum all the calculated sine values to get the result of the summation.