trig. value

[stuart clark]

can we Calculate value of \(\displaystyle \sin \frac{\pi}{7}\)

if yes than how can i calculate it

 

[mmm4444bot]

can we Calculate value of \(\displaystyle \sin \frac{\pi}{7}\)

It is possible, but the method is beyond introductory trigonometry because it involves expressing sin(Pi/7) using Complex exponentials, which can be difficult to simplify.

In other words, you cannot use elementary trigonometric identities to obtain an expression for sin(Pi/7) that involves only a combination of finite roots, sums, or products.

If you're interested in learning more about recognizing the values of n for which we can easily express the value of sin(Pi/n), then you could start by reading about constructible polygons or Fermat Prime numbers. If a regular n-gon is constructible, then sin(Pi/n) can be expressed using finite roots, sums, or products. These same values of n are also the Fermat Primes.

I think that Gauss was still a child when he demonstrated these values of n. Here's the first several:

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, …

 

[pka]

can we Calculate value of \(\displaystyle \sin \frac{\pi}{7}\)
if yes than how can i calculate it

If you have a calculator that will do sums easily then
\(\displaystyle \sum\limits_{k = 0}^4 {\left( { - 1} \right)^k \left( {\frac{\pi }{7}} \right)^{2k + 1} \left( {\frac{1}{{\left( {2k + 1} \right)!}}} \right)}\)
will give the answer correct to nine or ten places.

Note that is only five terms in the sum.