Heuristic method for solving $\sin^6 x+\cos^6 x=\dfrac{1}{4}$:
Method III:
Have you ever amazed and marveled by the powerful heuristic skill (the use of substitution, like we used in method I in previous post(Vietnamese Mathematical Olympiad (Trigonometric) Problem of 1962 )) that aimed to help in trick our mind (in a good way) to see the whole problem more clearly and so we could simplify things further to ease our work?
You would be even more staggered by what I am going to reveal to you on this blog post, I will show you how we could develop an already helpful and pretty heuristic thinking!
Showing posts with label cosine. Show all posts
Showing posts with label cosine. Show all posts
Vietnamese Mathematical Olympiad (Trigonometric) Problem of 1962
Solve the equation $\sin^6 x+\cos^6 x=\dfrac{1}{4}$.
This is one of the brilliant Mathematics Olympiad Contest Problems because we can show to the students how there are plenty of ways to attacking a good problem and how one approach is different from the other and how heuristic skill enable us to find solution quickly that save us time for more challenging problems!
This is one of the brilliant Mathematics Olympiad Contest Problems because we can show to the students how there are plenty of ways to attacking a good problem and how one approach is different from the other and how heuristic skill enable us to find solution quickly that save us time for more challenging problems!
A Trigonometric Sum
It can be shown that the following sum:
[MATH]S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right)[/MATH]
is rational. Find the value of $S$.
[MATH]S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right)[/MATH]
is rational. Find the value of $S$.
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