Articles of trigonometry

Help finding solution for trigonometric equation

I have a flat mirror and a target. Given the sun’s light angle of incidence, I must calculate how much to spin my mirror to hit the target with the light. The problem is shown in the following figure. Given $a$, the angle of sun’s light incidence, and $b$, the angle the reflected light must […]

Constructing a new function

I’ve got two function $f(x)=\sin(x)$ and $g(x)=\sin\left(\frac{x}{4}\right)$. I’d like to construct a function $h$, which would equal $0$ when $\sin(x)=0$ except when $\sin\left(\frac{x}{4}\right)=0$. So for $f(\pi)=0$ and $g(\pi)\ne0$ we have $h(\pi)=0$; for $f(2\pi)=0$ and $g(2\pi)\ne0$ we have $h(2\pi)=0$; for $f(3\pi)=0$ and $g(3\pi)\ne0$ we have $h(3\pi)=0$; for $f(4\pi)=0$ and $g(4\pi)=0$ we have $h(4\pi)\ne0$ (exact value doesn’t […]

What's wrong with this proof that $e^{i\theta} = e^{-i\theta}$?

I recently learned that $\cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin{\theta} = \frac{e^{i\theta} – e^{-i\theta}}{2}$ Based on this, I managed to “prove” that: $$e^{i\theta} = e^{-i\theta}$$ Since $e^{i\theta} = \cos{\theta} + i\sin{\theta}$, we can substitute the above two identities to get: $$e^{i\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2} + i\frac{e^{i\theta} – e^{-i\theta}}{2}$$ Simplifying, I get $$(1-i)e^{i\theta} = […]

Measuring diaognals without Sine Law

Lets start off with a simple right angled triangle ‘abc’. (ie: use cartesian coordinates, we mark ‘a’ and ‘b’ on x,y axis, ‘c’ is calculated from Pythagoras therom). Now pick an arbitrary point ‘o’ to the right of ‘c’. We then measure lengths e and f. Question: Can ‘q’ be calculated without using the Sine […]

Trig substitution for a triple integral

This problem involves calculating the triple integral of the following fraction, first with respect to $p$: $$ \int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} \frac{p^2\sin(\phi)}{\sqrt{p^2 + 3}} dp d\phi d\theta $$ This involves trig-substitution (I believe), and I am just hoping for an explanation of how to do it.

Calculating point on a circle, given an offset?

I have what seemed like a very simple issue, but I just cannot figure it out. I have the following circles around a common point: The Green and Blue circles represent circles that orbit the center point. I have been able to calculate the distance/radius from the point to the individual circles, but I am […]

Proof of vector addition formula

Two vectors of lengths $a$ and $b$ make an angle $\theta$ with each other when placed tail to tail. Show that the magnitude of their resultant is : $$r = \sqrt{ a^2 + b^2 +2ab\cos(\theta)}.$$ I understand that if we placed the two vectors head-to-tail instead of tail-to-tail, the Law of Cosines dictates that the […]

Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y

I recently came across the interesting definition of the sine function as the unique solution to the Initial Value Problem $$y” = -y$$ $$y(0) = 0, y'(0) = 1$$ (My first question would be why this solution is unique, but I found that proofs of uniqueness and existence are too complicated for me to understand) […]

Problem in the solution of a trigonometric equation $\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$

I needed to solve the following equation: $$\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$$ Now, the steps that I followed were as follows. Transform the LHS first: $$\begin{split} \tan\theta + \tan 2\theta+\tan 3\theta &= (\tan\theta + \tan 2\theta) + \dfrac{\tan\theta + \tan 2\theta} {1-\tan\theta\tan2\theta} \\ &= \dfrac{(\tan\theta + \tan 2\theta)(2-\tan\theta\tan2\theta)} {1-\tan\theta\tan2\theta} \end{split}$$ And, RHS yields $$\begin{split} \tan\theta\tan2\theta\tan3\theta […]

Calculate $x$, if $y = a \cdot \sin{}+d$

I am not an expert when it comes to trigonometric functions. I need to calculate value of $x$ for a program. If $y = a \cdot \sin{[b(x-c)]}+d$ then what will be the formula to calculate $x$? Please help..