Articles of trigonometry

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $2, 3, 4 $ are sufficient to express the general solution, for arbitrary degree equations, the general solution can be expressed only in terms of much […]

Find all different integer exponents

Find all different integers that satisfy the following equality: $m(\sin^{n}x + \cos^{n} x- 1) = n(\sin^{m}x + \cos^{m}x – 1), (\forall) x\in\mathbb{R}.$ Case1: $m$ is odd, $n$ is even, then put $x=180^0 => m=n=0 =>$ contradiction. Case2: $m$ and $n$ is odd, then put $x=180^0 => m=n =>$ contradiction. Case3: $m$ and $n$ is even […]

Confusing Trigonometry Problem

Lets say at an intersection the words “STOP HERE” are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle subtended by the letters to the eyes of a driver 20m from the base of the letters and […]

How to prove that $\cos\theta$ is even without using unit circle?

The proofs I have come across on showing that $\cos \theta$ is even is something like this: In a unit circle, $\cos\theta$ gives you the $x$ coordinate after traveling $\theta$ radians counterclockwise. Since, moving $\theta$ radians counterclockwise and $\theta$ radians clockwise i.e $-\theta$ will give you the same x coordinate, we have: $\cos(\theta)=\cos (-\theta)$ It […]

Longest pipe that fits around a corner.

This question already has an answer here: Intuitive explanation for formula of maximum length of a pipe moving around a corner? 3 answers

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+…..+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using the product-to-sum formula for $\cos(x)+\cos(y)$?

Finding a closed form for $\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$

This question already has an answer here: $\sum \cos$ when angles are in arithmetic progression [duplicate] 1 answer

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the Roll and it is \begin{pmatrix} cos(Pitch) sin(Yaw)\\ cos(Yaw) cos(Pitch)\\ sin(Pitch) \end{pmatrix}. How should it […]

Fixed point iteration convergence of $\sin(x)$ in Java

This question already has an answer here: Convergence of the fixed point iteration for sin(x) 1 answer

Evaluating $\frac{\sum_{k=0}^6 \csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$

The question is to evaluate $$\frac{\sum_{k=0}^{6}\csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$$ where $a=\pi/8$ without looking at the trigonometric table. I tried to transform the $\csc^2$ term to $\cot^2$ term and use addition formula but it made the problem too cumbersome.I also tried to manipulate the numerator in the form so that it telescopes but couldnot succeed.I am not in need […]