Articles of trigonometry

Riemann sum formulas for $\text{acos}(x)$, $\text{asin}(x)$ and $\text{atan}(x)$

In this post just another $\pi$ formula, I gave a kind of Riemann sum to compute the area of a quarter of circle based on a very simple geometric trick, and same reasoning can be used to compute any inverse trigonometric formulas based on the angle area. But I think that because of the simplicity […]

Finding sine of an angle in degrees without $\pi$

The following is found using a combination of: (a) a polygon with an infinite number of sides is a circle, (b) the perimeter of that polygon is the circumference of the circle that it becomes (of course), (c) the sine theorem, and (d) the ratio between the circumference of a circle and the diameter is […]

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can’t figure out why the values of $y$ take on the shape of straight lines and not curves looking more like those of sin, cos… EDIT: I understand that the derivative of Arccos(Sin(x)) gives 1 or -1 […]

conjecture regarding the cosine fixed point

context/motivation if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges – simply keep pressing the $cos$ button! this unique fixed point $\alpha$ might reasonably be expected to be a transcendental number. (perhaps the answer to that is […]

Equilateral Triangle Problem With Trig

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle P. The distance $|AP|=3 cm, |BP|=4 cm, |CP|=5 cm.$ What is the area of the triangle? I have seen this problem posted here before and solved, but done without trigonometry. How would […]

$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$

Prove that (not use induction) $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) \cos\big(\lfloor\frac{n+2}{2}\rfloor x\big)} {\sin x\cos^n x} \qquad\qquad (\frac{2x}{\pi}\not\in \mathbb Z)$

Equation of sine wave around a circle

Consider a sine wave having $4$ cycles wrapped around a circle of radius 1 unit (its center needs not be the origin of a Cartesian coordinate system). Assume that the length of axis of the sine wave is as same as the circumference of the circle. The circumference of circle is assumed to be mapped […]

How to solve the following summation problem?

$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$ Find value of $n$ for which equation is satisfied.

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My work: (a)I draw a chord $AB$ on the cirlce $C$, and choose any random point $P$ to […]

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at large times, the graph of $\sin$ oscillates very rapidly and so you can take it to be zero. When […]