Articles of trigonometry

Find the angle of depression

I am having trouble solving this word problem: A cellular tower that is 150ft is placed on top of a mountain that is 1200 feet above sea level. what is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above […]

Value of $ \cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $?

I am a little weak in trigonometry. I have two questions: Find the value of $\cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $ Find the value of $\sin 28^{\circ}+ \cos 17^{\circ} + \cos 28^{\circ} + \sin 17^{\circ} $ I am asking these questions because: 1. I am weak and unable to solve these. 2. […]

Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

I’m a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose roots was sin(80º) — which led me to the startling discovery that this (and, indeed, all rational values of sine/cosine (in […]

Apparent inconsistency between integral table and integration using trigonometric identity

According to my textbook: $$\int_{-L}^{L} \cos\frac{n \pi x}{L} \cos\frac{m \pi x}{L} dx = \begin{cases} 0 & \mbox{if } n \neq m \\ L & \mbox{if } n = m \neq 0 \\ 2L& \mbox{if } n = m = 0 \end{cases} $$ According to the trig identity given on this cheat sheet: $$ \cos{\alpha}\cos{\beta} = […]

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$. My try, considering $f(t)=\frac{\tan{x}}{\tan{y}}-\frac{x}{y}$ and derivating it to see whether the function is increasing in the given interval. I should be sure that $\lim_{x,y\rightarrow0}\frac{\tan{x}}{\tan{y}}-\frac{x}{y}\geq0$ for the previous derivative check to be useful, which I’m not yet, but I’m assuming it’s $0$ since I’d say that since both $x,y$ approach […]

value of $\tan(A)$

What is value of $tanA$ if $2\tan(2A)+4\tan(4A)+\frac{8}{\tan(8A)}=0$ writing everything in $\tan(A)$ and solving for $t$ is next to impossible without maths engines. So i am seeking for a shorter way. Thanks

Regular polygons and Pythagoras

Let $L_n:$ the side length of a regular $n$-polygon inscribed in a unit fixed circle. We have an interesting relationship: $L_6^2+L_6^2=L_4^2$ $L_6^2+L_4^2=L_3^2$ $L_{10}^2+L_6^2=L_5^2$ There are more solutions: $L_m^2+L_n^2=L_p^2$ ?

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):