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 […]

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 […]

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 […]

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 […]

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 […]

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

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 […]

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

$AB$ is a chord of a circle $C$. (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 […]

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 […]

Intereting Posts

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What is difference between cycle, path and circuit in Graph Theory
Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} – \det M_{ij}\det M_{ji}$
Calculating $\sum_{k=0}^{n}\sin(k\theta)$
Integrating each side of an equation w.r.t. to a different variable?
Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$
taylor series of ln(1+x)?
Why is $-\gamma = \int_0^1 \frac{e^{-z}-1}{z}dz+\int_1^\infty \frac{e^{-z}}{z}dz$
Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?
Negating the Legendre's conjecture
Combinatorics for a 3-d rotating automaton
The spectrum of normal operators in $C^*$-algebras
Finding the shortest distance between two Parabolas