Trig Definition and 1000 Threads

I understand why certain inverse trig functions have two answers. Like for arcsin(0.5), it could be pi/6 or 5pi/6. I know both angles have the same sin value, that they're both on the same horizontal line on a graph of sin, I get all of that, but two questions about it: 1) In cases where...

I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I...

Good Morning As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel. I stumble on terms like: 1. sin(theta)cos(theta) - cos(theta)sin(theta) 2. Cos_squared +...

Using the identity's (1) (2) Gives, Why dose this elegant method work? Many thanks!

Why when proving trig identities, Do we assume that r = 1 from ## rcis\theta = r[\cos\theta + i\sin\theta]##? This makes me think that this is somehow it is related the unit circle. Note: I am trying to prove the ##cos3\theta## identity and am curious why we assume that the modulus is 1...

Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0## ##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0## ##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here. Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0## ##\sin(x)\cos(x) +...

I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...

The trig identities for adding trig functions can be seen: But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change? Analysis: If the amplitudes do show up on the RHS, we would expect them...

##\tiny{2.4.2}## Differentiate ##f(x)=x\cos{x}+2\tan{x}## Product Rule ##[-x\sin{x}+\cos{x}]+[2\sec^2]\implies \cos{x}-x\sin{x}+2\sec^2x## mostly just seeing how posting here works typos maybe suggestions what forum do I go to for tikz stuff

I was just checking this out the sin##\frac {A}{2}## property, in doing so i picked a Right-Angled triangle, say ##ABC##, with ##AB=5cm##, ##BC=4cm## and ##CA= 3cm##. From this i have, ##s=6cm## now substituting this into the formula, ##sin\frac {A}{2}##= ##\frac {1×3}{5×3}##=##\frac...

Find the values of tan-1(1+2i). We can use the fact: tan-1z = (i/2)log((i+z)/(i-z)). Then with substitutions we have (i/2)log((1+3i)/(-i-1)). Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)). Do we then just proceed to solve log(1+3i) and log(-1-i)? I'm just a little confused...

##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## Let ##u = \csc{x}## then ##-du = \csc{x}\cot{x}dx## So, ##\int \frac{\csc{x}\cot{x}}{1+\csc^2{x}}dx## ##-\int \frac{1}{1+u^2}du = -\arctan{u} + C## ##-\arctan{\csc{x}} + C## This answer was wrong. The actual answer involved fully simplifying...

This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled...

Evaluate the integral $I_4=\displaystyle\int_{-\pi}^{\pi}\sqrt{\frac{1+\cos{x}}{2}} \, dx $ ok offhand i think what is in the radical is trig identity but might be better way...

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When solving for x I get the angles 0, pi, pi/2 and 3pi/2. However, I thought I should reject the pi/2 and 3pi/2 values since they are not in the domain of sec^2(x). I am using the opens tax precalc book and their answer does not reject those two angles.

$\begin{array}{lll} I&=\displaystyle\int{\frac{dx}{x^2\sqrt{x^2-16}}} \quad x=4\sec\theta \quad dx=4\tan \theta\sec \theta \end{array}$ just seeing if I started with the right x and dx or is there better Mahalo

This was the question, The above solution is the one that I got originally by the question setters, Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand), As you can see that both these methods give different answers...

Came across this trig identity working another problem and I've never seen it before in my life. I don't need to prove it myself, necessarily, but I would really like to see a proof of it (my scouring of the internet has yielded no results). If someone more trigonometrically talented than myself...

$\tiny{8.aux.27}$ Simplify the expression $\dfrac{{\cos 2x\ }}{{\cos x-{\sin x\ }\ }} =\dfrac{{{\cos}^2 x-{{\sin}^2 x\ }\ }}{{\cos x\ }-{\sin x\ }} =\dfrac{({\cos x}-{\sin x})({\cos x}+{\sin x\ })}{{\cos x}-{\sin x}} =\cos x +\sin x$ ok spent an hour just to get this and still not sure suggestions?

$\begin{array}{rl} \textit{Find } \mu(x): &\mu(x) =\exp\left(\int \dfrac{1}{x}\,dx\right)=e^{\ln{x}}=x\\ \textit{multiply thru by x} &xy^\prime+y=3x\cos 2x\\ \textit{rewrite as } &(xy)'=3x\cos 2x \\ \textit{}integrate &xy=\int 3x\cos 2x \...

v=197 If $y=x \sin x,$ then $\dfrac{dy}{dx}=$ $a.\quad\sin{x}+\cos{x}$ $b.\quad\sin{x}+x\cos{x}$ $c.\quad\sin{x}+\cos{x}$ $d.\quad x(\sin{x}+\cos{x})$ $e.\quad x(\sin{x}-\cos{x})$ well just by looking at it because $dx(x) = 1$ elimanates all the options besides b $1\cdot \sin (x)+\cos (x)x$...

Hello, Periodic trigonometric functions, like sine and cosine, generally take an angle as input to produce an output. Functions do that: given an input they produce an output. Angles are numerically given by real numbers and can be expressed either in radians or degrees (just two different...

Hello. Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist. But can a quotient of the two acutally approach a certain value? lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...

How do I work this out? Can’t seem to get my head around it! Thanks

y = 2sin(x) -1≤ sin(x) ≤ 1 -2 ≤ 2sin(x) ≤ 2 so -2 and 2 are the max/min limits but the domain is -π < x ≤ π Do I find the values of x that outputs -2 and 2 and show that they are within the domain ?

This is my first attempt ...

Hello, If I wanted to verify tan(x)cos(x) = sin(x), what about when x is pi/2? LHS has a restricted domain so it can't equal sin(x). Does this equation only work with a restricted domain? Nothing in my textbook discusses that. Thank you

## \int_0 ^ {2 \pi} \frac {dx} {3 + cos (x)} ## las únicas formas que probé fueron, multiplicar por ## \frac{3-cos (x)}{3-cos (x)} ## pero no me gusta esto porque obtengo una expresión muy complicada. También recurrí a la sustitución ## t = tan (\frac {x} {2}) ## que me gusta bastante, pero...

At what value of α is the curve y=asin2π/λ (x+α) in phase with z=asin2π/λ(x)? My answer booklet says α=1−λx+nλ, but I keep getting α=nλ, where n=0,1,2... I have no clue how to get to the answer shown in the mark scheme. Any insight would be much appreciated!

Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)? Why does arcsin (sin x) = x? Can it be that trig functions and their inverse undo each other?

Author gave solution C = \sqrt{2}, ∝ = -pi/4 but plugging C = - \sqrt{2}, ∝ = -3pi/4 into cos(x+y) and leaving the x I get \sqrt{2}Cos(x+3pi/4) = sinx+cosx Is my answer valid as well?

See the attached image. Apostol gives cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ), in the middle of the picture, but previous info given does not state how he got this equation. To me it looks like he used the equation cos(x+y) = cosxcosy-sinxsiny

Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?

unsure

Hi, I was hoping to get some help on these five questions, I've been stuck on these and any help would be greatly appreciated! 1. Inflation rises and falls in a cyclical manner. If inflation is highest at 4.8% and lowest at 1.3%, what c for the equation y = a sin(k(x + d)) + c? 2. If the...

QUESTION: ----------- For the purposes of this problem, we will define the direction of Vehicle A's initial velocity as the positive direction: While driving on a road that is inclined at an angle of 10 degrees above the horizontal, Vehicle A and Vehicle B are in a head-on collision lasting...

Kindly assist with this question: Determine the general solutions cos(2x+20)=-cos(x-11)

Summary:: This is not a homework i just need clarification on how they got those number Any solution involving sine and cosine is my weakness when its advance or intermediate

Homework Statement:: I need to develop my instincts on when to use u-sub, integration-by-parts, trig substitution, etc. But, I need to read/see tons of problems actually being solved with these techniques to know which technique to apply quickly. Relevant Equations:: Sorry for the vague...

My latest attempt \begin{align*} \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} = \\ \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} \cdot \frac{\csc \theta - \cot \theta}{\csc \theta - \cot \theta} =\\ \frac{\sin \theta \csc \theta + \tan\theta \csc \theta - \sin...

I have a few questions and a request for an explanation. I worked this problem for a quite a while last night. I posted it here. https://math.stackexchange.com/questions/3547225/help-with-trig-sub-integral/3547229#3547229 The original problem is in the top left. Sorry that the negative...

I'd like to solve ##0 = \cos^2(t) - t\sin^2(t)## but it's been forever since I've done some trig and I'm real rusty. I've tried rewriting terms using identities such as ##\sin^2(t) = 1 - \cos^2(t)## but not getting anything helpful. Can I get a point in the right direction?

Hey I stumbled across sinh, cosh, and tanh when trying to solve for some variables and I've tried to figure out how to use them but I don't really understand their definitions.I know that normal sine, cosine, and tangents have to do with a unit circle with the angle theta at the origin and the...

I am going over review to cover some trig fundamentals. Am stuck and am looking for the right place to get help for a question.

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...

I took Algebra I, II, and precalculus in high school, but I graduated high school some time ago and would like to prepare for calculus before I go back to school. My question is if a textbook that covers both Algebra and Trigonometry will sufficiently prepare me for calculus (I plan on taking...

Homework Statement: This is not a homework question. I am trying to understand why we spend so much time studying trig identities. Homework Equations: As far as I understand, the two basic trig functions (sin and cos ) show the relationship between the sides of a right angle triangle in a...

Evaluate using a trig identity $$\displaystyle \int \dfrac{5}{x^2\sqrt{25-x^2}}\, dx$$ my first inclination to set $u=5\sin{x}$ then $du=5\cos{x}\, dx$ or $dx=\dfrac{du}{5\cos {x}}$