Articles of trigonometry

Rigorous synthetic geometry without Hilbert axiomatics

Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations? I am aware of Hilbert’s foundations and the […]

Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $

I’m trying to find $$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$ I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using De Movire’s, but so far nothing has come up.

Absolute value in trigonometric substitutions

In general, when we are trying to remove radicals from integrals, we perform a trigonometric substitution (either a circular or hyperbolic trig function), but often this results in a radical of the form $\sqrt{(f(x))^2}$, with $f$ being an arbitrary trigonometric function. What most texts tend to do is simply take $\sqrt{(f(x))^2} = f(x)$, without the […]

query about the cosine of an irrational multiple of an angle?

de Moivre’s identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will be $n$ values of $(\cos \theta + i \sin \theta)^{\frac{m}{n}}$, whilst for irrational $\alpha$ the set $\{e^{i\alpha (\theta +2\pi […]

How to solve $A\tan\theta-B\sin\theta=1$

I was wondering if it is possible to solve $$A\tan\theta-B\sin\theta=1$$ for $\theta$, where $A>0,B>0$ are real constants. For sure this can be straightforwardly implemented numerically, but maybe an alternative exists :)…

Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$

I am working on an integration by parts problem that, compared to the student solutions manual, my answer is pretty close. Could someone please point out where I went wrong? Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$ $u_1 = \sin{3\theta}$ $du_1 = \frac{1}{3}\cos{3\theta} \ d\theta$ $v_1 = \frac{1}{2} e^{2\theta}$ $dv_1 = e^{2\theta} \ d\theta$ $\underbrace{\sin{3\theta}}_{u_1} […]

Isosceles triangle

$\Delta ABC$ in the figure below: $\angle 1+\angle 2=\angle 3+\angle 4,\quad$ $E\in AB,\; D\in AC,\; F=BD\cap CE,$ $BD=CE$. Prove: $AB=AC$ The exact version figure should look like: This problem should be a little more difficult than the Steiner-Lehmus Theorem.

$(\tan^2(18^\circ))(\tan^2(54^\circ))$ is a rational number

Assuming $$\cos(36^\circ)=\frac{1}{4}+\frac{1}{4}\sqrt{5}$$ How to prove that $$\tan^2(18^\circ)\tan^2(54^\circ)$$ is a rational number? Thanks!

Rigorous proof of an infinite product.

I’ll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos \frac{x}{4}\sin \frac{x}{4}$$ $$ {\sin x} = 2^3 \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \sin\frac{x}{8} $$ $$ {\sin x} = 2^k \cos \frac{x}{2} \cos \frac{x}{4} \cdots\cos \frac{x}{2^k} \sin\frac{x}{2^k} $$ […]

Solving $\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$

I’m trying to solve $$\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$$ By substituting $x=2\tan{t}$. I get as far as: $$\int_{x \space = -\infty}^{x \space = \infty}{\frac{1}{(4+(\underbrace{2\tan{t}}_{x})^2)\sqrt{4+(\underbrace{2\tan{t}}_{x})^2}} \cdot \underbrace{2(1+\tan^2t) \space dt}_{dx}} = \dots$$ $$\dots = \frac{1}{4} \cdot \int_{t \space = -\infty}^{t \space = \infty}{\frac{1}{\sqrt{1+\tan^2t}} \space dt}$$ Now what? Have I done anything wrong? I don’t see how I could continue […]