Articles of trigonometry

When is $\sin x$ an algebraic number and when is it non-algebraic?

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 […]

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

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 […]

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

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

Cotangent summation (proof)

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

Minimum value of $2^{\sin^2x}+2^{\cos^2x}$

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?

Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$

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 $x^2 – 44x – 36 = 0$ holds for $x=\sin 2A$

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.

Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

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

What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?

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 […]

Solving trigonometric equations of the form $a\sin x + b\cos x = c$

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 […]