Trigonometric Definition and 1000 Threads

Hey everyone . So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking ! So i was...

This could be solved by the substitution ##u=\sqrt x##, but I wanted to do it using a trigonometric one. The answer is false, but I don't see the wrong step. Thank you for your time! [Poster has been reminded to learn to post their work using LaTeX]

Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...

Homework Statement: Solve for x. Homework Equations: sin(3x)= -1/2 sin(3x) = -1/2 3x = sin-1(-1/2) 3x = -π/6 x = -π/18 x = -π/18 + 2π/3 = 11π/18 11π/18 + 2π/3 = 23π/18 11π/18 + (2π(4))/3 = 35π/18 The solutions I obtained were 23π/18 and 35π/18. Are these correct? I'm not entirely sure...

Show that \[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\] - and deduce that \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]

theta1 = 90- theta2 I substituted that into v^2*sin(2theta1)/g So I get v^2*sin(180-2theta2)/g Now I'm stuck. What do I do next?

Hello everyone, I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...

We know the answer, but don't know how it makes sense given trigonometric principles. Borrowed from HiSet free practice test

In this question, I tried this: sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x), where cosec(x) = 1/sin(x) and sec(x) = 1/cos(x) -sin^2(180-x) = sin^2(x) and cos^2(x) = cos^2(x) -The sin^2 and the 1/sin(x) cancle out along with the cos^2 and the 1/cos(x) Therefore, I am left with...

This is the integral I try to take. ##\int\sqrt{1+9y^2}## and ##9y^2=tan^2\theta## so the integral becomes ##\int\sqrt{1+tan^2\theta}=\sqrt {sec^2\theta}##. Now I willl calculate dy. ## tan\theta=3y ## and ##y=\frac {tan\theta}3## and ##dy=\frac{1+tan^2\theta}3## Here is where I can only...

I can not understand why ##v_x = -|v|sin(θ)## and ##v_y = |v|cos(θ)## I'm asking about the θ angle. If i move the vector v with my mind to the origin i get that the angle between x'x and the vector in anti clock wise, it's 90+θ not just θ. So why is he using just θ? Does the minus in v_x somehow...

Homework Statement I need to solve a system of two equations for T and θ algebraic and with all the other parameters known. φ is equal to: Homework Equations The relevant equations are the two equations left of * in the image below The Attempt at a Solution I tried Gauss elimination but I...

Homework Statement The equation; 0.966= -0.354sin(φ+60) + 0.935cos(φ+60) and we're trying to find φ. Homework Equations 3. The Attempt at a Solution [/B] (edited) I tried doing ^2; 0.933= 0.125sin2(φ+60) -2*0.331sin(φ+60)cos(φ+60) + 0.874cos2(φ+60) x=φ+60 0.933= 0.125sin2x - 0.331sin(2x) +...

I already know the fact that angles are physical quantities, but sin, cos of some angles are quantities? Quantities are those things, which can be quantified, are sin, cos, tan be quantified through measurement, if yes then other mathematical functions should also be categorised as physical...

<Moderator's note: Moved from a technical forum and thus no template.> Mechanics by Lev D. Landau & E. M. Lifshitz Chapter 4 Collisions between particles §16. Disintegration of particles Problem 3 The angle θ = θ1 + θ2 It is simplest to calculate the tangent of θ. A consideration of the...

I have a trigonometric equation 2\sin \left ( \frac{q\pi }{m} \right )-\sin \left ( \frac{q\pi }{2} \right )=0 and want to know what values m as a function of q could take to satisfy the equation. Both terms zero is the obvious solution: q=2n; m=2; n is an integer. But there are more solutions...

The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain! - - - Updated - - - Oh nvm I see my mistake!

Find the smallest natural number $n$ for which $\sin \left(\dfrac{1}{n+1934}\right)<\dfrac{1}{1994}$.

Homework Statement Find the area bounded by arcsinx, arccosx and the x axis. Hint-you don't need to integrate arcsinx and arccosx Homework Equations All pertaining to calculus The Attempt at a Solution I drew the correct graph and marked their intersection at (1/√2, pi/4) and painstakingly...

Homework Statement Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1 Homework Equations All trigonometric and inverse trigonometric identities, special usage of double angle identities here The Attempt at a Solution I can get the answer by puting x=cosy, the term inside...

Homework Statement solve ##\int_0^1 x^6 \arcsin{x} dx##

Homework Statement If ## I_n = \int_0^\frac {\pi}{4} \sec^n x dx## then find ## I_{10} - \frac {8}{9} I_8## 2. The attempt at a solution this should be solvable by reduction formulae but since it'd be longer I wanted to know if there was a way to do it using mostly properties of indefinite...

Homework Statement Show that sec(x-(pi/2))+tan(x-(pi/2))=tan((x/2)+pi/12)) The Attempt at a Solution I applied all sorts of half angle formulas to convert it in terms of tan, I got LHS as (tan((x/2)-(pi/6))+1)^2/1-tan^2(x/2-(pi/6)) but I'm sure there must be a simple easy method to get the RHS...

Today,in our class, we received a trigonometric equation ##\sin^{10}{x}+\cos^{10}{x}=\frac{29}{16}\cos^4{2x}## Here is my attempt:

If xy+yz+zx=1 then prove (x^2-1)(y^2-1)/xy+(y^2-1)(z^2-1)/yz+(z^2-1)(x^2-1)/zx=4 with trigonometric identities such as cotAcotB+cotBcotC+cotCcotA=1

Homework Statement ##4y=cos\left(4πx+\frac{3}{2}\right)## Homework EquationsThe Attempt at a Solution In dividing both sides by 4, I got: ##y=\frac{1}{4}cos\left(πx+\frac{3}{8}\right)## But I am told this is incorrect. Not sure if dividing everything by 4 here is an allowable technique, or if...

The circle ## x^n+y^n=1 ##, for n integer >2 in a metric space with distance function: ## \sqrt[n] {dx^n+dy^n} ## has corresponding trigonometric Sine and Cosine functions defined in the usual way. Finding the sine or cosine of the sum of two angles, derivatives and curvature of a line in such...

Hello everyone Can someone help me out solving this integral: \begin{equation} S_T(\omega)=\frac{2k_BT^2g}{4\pi^2c^2}\int_0^{\infty}\frac{sin^2(kl)}{k^2l^2}\frac{k^2}{D^2k^4+\omega^2}dk \end{equation} Where $$D=g/c$$ According to this paper https://doi.org/10.1103/PhysRevB.13.556. The...

Homework Statement i have no idea how to proceed

Homework Statement Triangle ABC is shown in the diagram below. A C AC = 3AB <BAC = 120° Respectively <BCA = Show that angle BCA can be written in the form...

Prove, that $$\prod_{j = 1}^{n}\left(1+2\cos \left(\frac{3^j}{3^n+1}2\pi\right)\right) = 1.$$

Given h(x) = tan x, evaluate dh/dx on [pi/4, 1]. Note: d = delta I need one or two hints. I can then try on my own.

I am in the trigonometry section of my precalculus textbook by David Cohen. In Section 6.2, David explains how to evaluate trig functions without using a calculator but it is not clear to me. Sample: Is cos 3 positive or negative? How do I determine if cos 3 is positive or negative without...

Hello all, I am trying to solve the integral: \[\int cot(x)\cdot csc^{2}(x)\cdot dx\] If I use a substitution of u=cot(x), I get \[-\frac{1}{2}cot^{2}(x)+C\] which is the correct answer in the book, however, if I do this: \[\int \frac{cos(x)}{sin^{3}(x)}dx\] I get, using a substitution...

If 0 < x < 90, what is the solution of tan(2x - 5) = cot(x + 5)? I got stuck in tan(2x - 5)tan(x + 5) = 1. What should I do after that?

Homework Statement cos2x + cos x = 0 (0 <= x <= 360) Homework EquationsThe Attempt at a Solution cos2x + cos x = 0 2cos(3x)/2 cos(x)/2 = 0 3x/2 = 90 degrees x = 60 degrees x/2 = 90 x = 180 3x/2 = 270 x = 180 x/2 = 270 x = 540 (not qualified) is there any more possibility (answers) for x?

Homework Statement Just a simple problem, I need to take the expression##\frac 1 2 (sin(2x)+1)## and show it is equivalent to ##sin^2(x+\frac \pi 4)##, and I can't seem to manage to find the way to do so, so I would appreciate some insight. Homework Equations N/A The Attempt at a Solution...

Prove the identity \[\sum_{j=1}^{n-1}\csc^2\left ( \frac{j\pi}{n} \right ) = \frac{n^2-1}{3 }.\]

Homework Statement Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx Homework Equations cot(x) = 1/tan(x) sin^2(x) + cos^2(x) = 1 The Attempt at a Solution I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved...

Could I please get help with the following question? f(x)=(2cos^2 x+3)^5/2 Any help would be very much appreciated:)

$\tiny{2214.t1.14}$ $\text{Evaluate the Integral:}$ \begin{align*}\displaystyle I_{14}&=\int \frac{12\tan^2x \sec^2 x}{(4+\tan^3x)^2} \, dx \\ \textit{Use U substitution}&\\ u&=4+\tan^3x\\ \, \therefore dx& =\dfrac{1}{3\sec^2\left(x\right)\tan^2\left(x\right)}\,du\\ &=4...

Prove $$\sum^{(N-1)/2}_{n=1}\cos\left[\frac{\pi}{N}(2n-1)\right]=\frac12$$ For $N=3,5,7...$.

Prove $\sin 15^\circ \sin 24^\circ \sin 57^\circ= \sin 39^\circ \sin 27^\circ \sin 18^\circ$. This is an unsolved problem I found @ AOPS.

$\frac{1 -\cos A}{1 + \cos A} = (\cot A - \csc A)^2$

Consider the following set of equations: ##r = \cosh\rho \cos\tau + \sinh\rho \cos\varphi## ##rt = \cosh\rho \sin\tau## ##rl\phi = \sinh\rho \sin\varphi## Is there some way to combine the equations to get rid of ##\varphi## and ##\tau## and express ##\rho## in terms of ##r, t, \phi##? I...

Homework Statement Homework Equations The Attempt at a Solution Here is my answer, i get 1/24 For my first step i divided both terms under the radical by 4, then split 1/4 into (1/2)2, i saw something very similar in my book so i did the same thing, but i just realized this has to be...

Show, that $5x \le 8\sin x - \sin 2x \le 6x$ for $0 \le x \le \frac{\pi}{3}$.

I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...

Evaluate: \[\lim_{x\rightarrow 0}\frac{\sin (\arctan x)-\tan (\arcsin x)}{\arcsin (\tan x)-\arctan (\sin x)}\]