Questions tagged [trigonometry]
Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.
30,538 questions
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The area of a rectangle containing a curved line of a known length
Let's suppose we have a line with a known length $L$ that we transform from a straight line to a curved one matching a semi-circle.
Is there a way to find a formula for the height of the smallest ...
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Solving $\int{\frac{dx}{\sin(x)\cos(x)}}$ using $\tan\left(\frac{x}{2}\right)$ substitution
I have the following indefinite integral (Primitive) in the interval $\left(-\frac{\pi}{2},0\right)$:
$$\int{\frac{dx}{\sin(x)\cos(x)}}$$
I wanted to try another method to solve the above integral, ...
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Prove $\frac{\gamma^2 - \alpha^2 - \beta^2}{2\alpha\beta}\ge\frac{c^2-a^2-b^2}{2ab}$
Given three positive numbers $a,b,c$ such that $c\ge a+b$.
For three positive numbers $x,y,z$
constrained by the triangle inequalities for each set of sides $(x,z,a)$, $(x,y,b)$, and $(y,z,c)$:
$|x - ...
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On proving $\cos\left(\frac{1}{3}\arccos\left(\sqrt{\frac{27x}{2(2x+3)^3}}\right)-120^\circ\right)+\sqrt{\frac{6x}{(2x+3)(4x+3)^2}}>0$
Given a positive $x$ we want to prove:
$$\cos\left(\frac{1}{3}\arccos\left(\sqrt{\frac{27x}{2(2x+3)^3}}\right)-120^\circ\right)+\sqrt{\frac{6x}{(2x+3)(4x+3)^2}}>0$$
The plot of the left hand side ...
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Prove that $\tan^{-1}(x) = -\tan^{-1}(-x)$
Prove that $\tan^{-1}(x) = -\tan^{-1}(-x)$.
I was looking for this proof online and could not find it for some reason. I did find this video for the proof, but YouTube said the video had been taken ...
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Deciphering some handwritten equations from Ramanujan's Second Notebook
As shown, on pg 259 of Ramanujan's 2nd notebook he states a few identities:
I am easily able to decipher the second half, which just consists of some class equations for the theory of alternative ...
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Better and Exact Solution on The Nature of n At the Limiting [closed]
Consider a right angled triangle ABC right angled at B, with sides AB, BC, and AC as $\sqrt{n}$, 1 and $\sqrt{n+1}$ units respectively. Also, an angle theta is present between the sides AC and BC. The ...
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The Trigonometric Ratios of Angles more than $90^\circ$ [closed]
I understood the whole concept, but I had to make this assumption to understand it, Here
Is this Correct understanding of the topic ?
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Let $f(x) R^{+} \rightarrow R^{+}$ be a triangle preserving function.Prove that there exist A,B such that $f(x)\leq Ax+B$ for all $x\in R^{+}$
Let $f(x) R^{+} \rightarrow R^{+}$ be a triangle preserving function(if $a,b,c \in R^{+}$ are side lengths of some triangle, then $f(a),f(b),f(c)$ are, too.)Prove that there exist A,B such that $f(x)\...
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Understand the plot of inverse trigonometric functions [duplicate]
I am a student of Mathematics and while preparing for my exam, I sought to think of a shortcut to plot all 6 inverse trigonometric functions. I obtained the below steps. On paper with hand drawn ...
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{Clothoid Navigaion} projecting a startpoint and heading to a clothoid-interpreted point(p2) and turn radius
Just a reminder that I'm not a mathematician so apologies for any syntactical inaccuracies. Also, please know this was NOT generated by, nor will be created FOR anything AI related. This is simply ...
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How can I accelerate the convergence rate of trigonometric and inverse trigonometric functions?
I was expelled from high-school and I slept through all classes, I never went to university so I genuinely don't know how to accelerate the convergence rate. I searched everywhere using Google and I ...
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Conjecture: Precise generalization of triangle inequality for cyclic polygons
Motivation: In any convex polygon, the length of any side is less than the sum of all the other sides but we don't know is by how much. So I am trying to obtain precise generalization of the triangle ...
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Help understanding inverse kinematics equations of 5 bar linkage
I am trying to understand the equations for an inverse kinematics linkage system I found in a documentation. I already have the solutions however I can't get my head around how these work.
Linkage ...
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How to simplify an expression involving trigonometric functions into its simplest form
I apologize for probably asking the same questions over and over again, but I’d like to put the pieces of my mental puzzle together.
Consider the following expressions:
$$
sc(b)=\sum_{n=1}^{x} \frac{\...
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