Articles of trigonometry

High School Projectile Motion and Quadratics

High school students are learning about the basics of solving quadratics and trigonometric ratios, including trigonometric inverses. The eventual goal of their project is to be able to show a reasonable firing solution, given in initial angle $\theta$ and initial velocity $v_0$. Projectile motion is given by $$y=\left(\tan{\theta}\right)x-\left(\frac{g}{2v_0^2\cos^2{\theta}}\right)x^2$$ where $x$ and $y$ are horizontal and […]

Solving the ArcTan of an angle (Radians) by hand?

How do you solve $\arctan(n)$ to radians by hand? I. e. $\arctan(1)$ >> process >> $\pi/4$ ::EDIT:: I have this taylor expansion that allows me to calculate an approximate value for arctan, but am wondering if there’s a closed-form solution (Or a more general formula than below):

Range of a 1-2 function

$$f(x)=\frac x {x^2+1}$$ I want to find range of $f(x)$ and I do like below . If someone has different Idea please Hint me . Thanks in advanced . This is 1-1 function $\\f(x)=\dfrac{ax+b}{cx+d}\\$, This is 2-2 function $\\f(x)=\dfrac{ax^2+bx+c}{a’x^2+b’x+c’}\\$, This is 1-2 function $\\f(x)=\dfrac{ax+b}{a’x^2+b’x+c’}\\$

Drawing sine and cosine waves

I like mathematics and pretty much every mathematical subject, but if there is one thing I thoroughly dislike, it is drawing (functions, waves, diagrams, etc.) We have this important trig test coming up and I need to master the drawing of sine and cosine waves. Can you guys give me an action plan of how […]

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 […]