Evaluate a floor function involving trigonometric functions

In summary, the trinomial in the expression is a perfect square and it provides ideas for solving the problem.
  • #1
anemone
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MHB
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Evaluate \(\displaystyle \left\lfloor{\tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right)}\right\rfloor\).

Hi MHB,

I don't know how to solve the above problem, as I have exhausted all possible methods that I could think of, and I firmly believe there got to be an easy way to crack it because this is a competition problem...any help, please?:)
 
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  • #2
Re: Evaluate a floor function involves of trigonometric functions

Hmm. That is, indeed, a tough one. The answer, according to my calculator, is $412$, but how to get that? I'm thinking esoteric trig identities are the way to go. Here's one that might be useful:
$$\cos\left(\frac{\pi}{7}\right)\cos\left(\frac{2\pi}{7}\right)\cos\left(\frac{3\pi}{7}\right)=\frac18.$$
Here's another:
$$\prod_{k=1}^{m}\tan\left(\frac{k\pi}{2m+1}\right)=\sqrt{2m+1}.$$
Fleshing this out for your case yields
$$\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)=\sqrt{7}.$$
We can combine these two together to get
$$\sin\left(\frac{\pi}{7}\right)\sin\left(\frac{2\pi}{7}\right)\sin\left(\frac{3\pi}{7}\right)=\frac{\sqrt{7}}{8}.$$
Hmm. Squaring some of your expressions looks like we might be able to do something here.

I also noticed that there's a perfect square trinomial pattern hidden in your original expression:
\begin{align*}
&\left\lfloor\tan^4\left(\frac{3\pi}{7}\right)+\tan^4\left(\frac{2\pi}{7}\right)+2\left(\tan^2\left(\frac{3\pi}{7}\right)+\tan^2\left(\frac{2\pi}{7}\right)\right)\right\rfloor \\
=&\left\lfloor\left(\tan^2\left(\frac{3\pi}{7}\right)+1\right)^{\!2}+\left(\tan^2\left(\frac{2\pi}{7}\right)+1\right)^{\!2}
-2\right\rfloor \\
=&\left\lfloor \sec^4\left(\frac{3\pi}{7}\right)+\sec^4\left(\frac{2\pi}{7}\right)-2\right\rfloor.\end{align*}

I'm not sure where to go from here; does this give you any ideas?
 
  • #3
Thanks so much Ackbach for your reply!

I will think of it based on your observations and hopefully I can crack it soon and when I have done so, I sure will post back...it may take a while as I am very, very busy these days...
 

FAQ: Evaluate a floor function involving trigonometric functions

1. What is a floor function?

A floor function, denoted by the symbol ⌊x⌋, is a mathematical function that takes a real number as an input and returns the largest integer less than or equal to that number. In other words, it rounds down the input to the nearest integer.

2. How is a floor function evaluated?

Involving trigonometric functions, a floor function can be evaluated by first solving the trigonometric expression inside the function, and then rounding down the resulting value to the nearest integer. For example, if the expression inside the floor function is sin(π/4), the result would be 0. The floor function would then round this down to 0, making the final result ⌊0⌋ = 0.

3. Can a floor function involve multiple trigonometric functions?

Yes, a floor function can involve multiple trigonometric functions. In this case, each trigonometric expression should be solved first, and then the resulting values should be rounded down separately. The final result would be the floor function of the sum of the individual results.

4. What is the purpose of a floor function in mathematical calculations?

The floor function is often used in mathematical calculations to round down the result of a calculation to the nearest integer. This can be useful in various applications, such as in computer programming, where only integer values can be stored in certain data types.

5. Are there any special cases to consider when evaluating a floor function with trigonometric functions?

Yes, there are a few special cases to consider when evaluating a floor function with trigonometric functions. For example, if the input to the floor function is negative, the floor function would round down to the next integer in the negative direction. Additionally, if the input is an integer, the floor function would return the same integer as the output.

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