Show that if $x$ is rational, then $\sin x$ is algebraic number when $x$ is in degrees and $\sin x$ is non algebraic when $x$ is in radians. Details: so we have $\sin(p/q)$ is algebraic when $p/q$ is in degrees, that is what my book says. of course $\sin (30^{\circ})$, $\sin 45^{\circ}$, $\sin 90^{\circ}$, and […]

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and using the strong induction principle. (Problem on Pg:84, 70 in typeset; solution on Pg:126, 111 in Typeset). However, for Completeness, I’ll […]

(source for above graph) (source for above graph) Both functions simplify to x, but why aren’t the graphs the same?

How to sum up this thing, i tried it with complex number getting nowhere so please help me with this,$$\sum_{k=0}^{n-1}\cot\left(x+\frac{k\pi}{n}\right)=n\cot(nx)$$

The question is what is the minimum value of $$2^{\sin^2x}+2^{\cos^2x}$$ I think if I put $x=\frac\pi4$ then I get a minimum of $2\sqrt2$. But how do I prove this?

I have two similar looking questions. $(1)$Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$ $(2)$If $\Delta ABC$ is an acute angled,then prove that $\cos^2A+\cos^2B+\cos^2C<\frac{3}{2}$ If I apply Jensen’s inequality, then $\cos^2x$ is a concave function, because its second derivative is $-2\cos 2x$ and with it being concave function $\cos^2A+\cos^2B+\cos^2C\leq\frac{3}{4}$ which is not there in the question.How will we […]

If $$\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$$ then prove that $$\sin 2A \quad\text{ is a root of }\quad x^2 – 44x – 36 = 0$$ I have no idea how to solve it. Plz help.

This question already has an answer here: Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$ 2 answers

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ $$\big(\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}\big)^3 = \frac{5-3\cdot 7^{1/3}}{2}\tag{2}$$ $$\big(\cos(\tfrac{2\pi}{11})^{1/5}+\cos(\tfrac{4\pi}{11})^{1/5}+\dots+\cos(\tfrac{10\pi}{11})^{1/5}\big)^5 = x?\tag{3}$$ Question: What degree is the minimal polynomial of $x$? Since the previous two are deg 2 and 3, I had assumed (3) would be deg 5, but Mathematica does not recognize it as a quintic with small coefficients, nor a 25th deg (even […]

Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0<x<2\pi$. How do you solve this equation without using the method that moves […]

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