# Tagged Questions

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### Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
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### Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
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### Simple Trigonometry and algebra

If $$\sec\theta = X + \frac{1}{4X},$$ then what is $${\sec\theta + \tan\theta}$$ in terms of $X$?
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### Method for finding roots of real trigonmetric polynomial

Given a real valued trigonometric polynomial, $$f(x) = \sum_{k=0}^{n} a_k \cos(k x + \phi_k)$$ what is the current fastest numerical method to find the roots of the polynomial for a given error? I ...
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### Help me solve this olympiad challenge?

Given: $$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$ What is the sum of all arctan of all the roots of $p(x)$?
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### How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
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### If z is one of the fifth roots of unity, not 1…

If z is one of the fifth roots of unity, not 1, show that: $1+z+z^2+z^3+z^4=0$ Which wasn't too bad, but the next part is killing me: show that: $z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$ Can ...
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### Changing the argument for a higher order derivative

I start with the following: $$\frac{d^n}{dx^n} \left[(1-x^2)^{n+\alpha-1/2}\right]$$ Which is part of the Rodrigues definition of a Gegenbauer polynomial. Gegenbauer polynomials are also useful in ...
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### rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
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### How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
I've been thinking a bit about finding the minimal polynomials of side lengths of regular $n$-gons inscribed in the unit circle. For example, I recently wanted to find the minimal polynomial of the ...
The problem is to figure out how to solve a real cubic equation of the form $x^3 + px + q = 0$ using trigonometry. The first step is to prove the identity  4\cos^3 \theta - 3\cos \theta - \cos ...