Articles of trigonometry

If $\frac{\sin A}{\sin B}=\frac{\sqrt{3}}{2}$ and $\frac{\cos A}{\cos B}=\frac{\sqrt{5}}{2}$, then $\tan A+\tan B = $???

$$\text{If}\quad \frac{\sin A}{\sin B}=\frac{\sqrt{3}}{2} \quad\text{and}\quad \frac{\cos A}{\cos B}=\frac{\sqrt{5}}{2}\,, \quad\text{then}\quad \tan A+\tan B = \text{???}$$ Here, $0<A,B<\frac\pi2$. I tried many times but I am getting no result. Please help with some hint.

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = \omega^3-p_2\omega+p_6\sin(3\omega\tau)$. Forget about all the other parameters/variables in the equations. All I want is to find a closed form expression for $\tau$. Thanks for your help and ideas!

How can I solve for a single variable which occurs in multiple trigonometric functions in an equation?

This is a pretty dumb question, but it’s been a while since I had to do math like this and it’s escaping me at the moment (actually, I’m not sure I ever knew how to do this. I remember the basic trigonometric identities, but not anything like this). I have a simple equation of one […]

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, I was informed by Joriki that $$\int \cos\left(\frac{1}{x}\right) \mathrm{d}x = x \cos\left(\frac{1}{x}\right) + \operatorname{Si}\left(\frac{1}{x}\right)$$ where $$\mbox{Si}(u) = \int \frac{\sin(u)}{u} […]

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn’t solve it so I’m posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} \sin\dfrac{2\pi\cdot j^2}{n}$$ Then prove that if $n \equiv 0,3 \pmod{4}$ we have $S = \frac{\sqrt{n}}{2}$ and if $n \equiv […]

Simplifying trig expression for Laplace transform

I’m working on the following Laplace transform problem at the moment, and I’m a little stuck. $$\mathcal{L} \{\sin(2x)\cos(5x) \}$$ I don’t recall any trig identity that would apply here. I know that $$\sin(2x) = 2\sin(x)\cos(x)$$ But I’m not sure if that applies in this situation. If you guys could point in the right direction I’d […]

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I’ve learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is true whenever the argument of sine matches the denominator and they both tend to zero. That is, $$\lim_{x \rightarrow 0}{\frac{\sin(5x)}{5x}} = […]

Tan Binomial formulas from a set S and its k-subsets

Working around, I found some Tan Binomial formulas. Let’s $S$ be a set such that: $$ S=\left\{\text{ }\tan ^2\left(\frac{1\pi }{n}\right), \tan^2\left(\frac{2\pi }{n}\right), \tan^2\left(\frac{3\pi }{n}\right)\text{ },\text{…},\tan^2\left(\frac{k \pi }{n}\right) \right\} \text{for } k \text{ in } \text{range } \left[ 1,\left\lfloor \frac{(n-1)}{2}\right\rfloor \right] $$ and let’s $S_k$ be a k-subset of $S$. For example, for k=2, we have: […]

Spread Polynomials Identity (Rational Trigonometry)

Show that if $ n=2l+1 $ is an odd natural number then $$ S_{n}\left( s \right)=s\left( \binom{n}{1}\left( 1-s \right)^{l}-\binom{n}{3}\left( 1-s \right)^{l-1}s+\binom{n}{5}\left( 1-s \right)^{l-2}s^{2}-\cdots+(-1)^{l}s^{l} \right)^{2} $$ (Exercise 8.7 on page 107 of N J Wildberger’s book Divine Proportions.) First let us review our tools: Define the Spread Polynomials by a recursive relation:$$ S_{n+1}\left( s […]

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?