Trigonometric Definition and 1000 Threads

A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to $\theta$ when $\theta...

I need to find the domain of this function: $$f(x) = \frac{1+x}{ e^{cos(x)}}$$ I set $${ e^{cos(x)}}> 0$$ But I'm not sure what to do after this.

Transform the left hand member into the right hand member. $\frac{\tan\alpha+\tan\beta}{\sec\alpha-\sec\beta}=\frac{\sec\alpha+\sec\beta}{\tan\alpha-\tan\beta}$By using cross multiplication I was able to prove this identity but what I actually want to accomplush is to transform the left member...

I already did everything that I can to transform the left side member to the right side member but I always get a jumbled terms. Please give me a hand on this problem. $(2\sin^{2}(\theta)-\cos^{2}(\theta))^{2}-9(2\sin^{2}(\theta)-1)^{2}=(2-3\sin^{2}(\theta))(2+3\sin(\theta))(3\sin(\theta)-2)$

Mod note: Moved from a technical math forum, so this post is missing the homework template i am trying to prove that ##1/sec∅-tan∅ ≡ sec ∅ + tan∅## this is how i attempted it, i tried to show that the left hand side is equal to the right... ## 1/ 1/(cos∅-sin∅)/cos∅## where i end up with ##...

transform the first member to the second member $\sqrt{\frac{\sec^{2}(\phi)-1}{\sec^{2}(\phi)(1+\cot^{2}(\phi))}}+\frac{\csc^{2}(\phi)}{\csc(\phi)}\sqrt{\frac{\csc^{2}(\phi)-1}{\csc^{2}(\phi)}} = (1+\cot(\phi))(1-\sin(\phi)\cos(\phi))$this is what I tried so far...

Help me get started with these problems. Prove the following$\frac{\tan(A)}{1-\cot(A)}+\frac{\cot(A)}{1-\tan(A)}=\sec(A)\csc(A)+1$ $\frac{\sec(A)-\tan(A)}{\sec(A)+\tan(A)}=1-2\sec(A)\tan(A)+2\tan^{2}(A)$

Homework Statement Hi there, I am attempting to prove a trigonometric equation using the half angle and double angle formulae Homework Equations See image one... The Attempt at a Solution See image two... I get stuck after the second line and can't see how to continue, please help.[/B]

Find the value of x between 0 degrees and 360 degree satisfying the equation 10sin^2x+ 10sin x cos x - cos^2x = 2 this is how i have attempted..... 10 sin^x+ 10sin2x/2 - cos^x = 2 I used the property sin 2x = 2 sin x cos x and substituted sin x cos x with sin 2x/2 giving me.... 11sin^2x +...

I'm solving two different definite integrals of functions \frac{sin(z)}{z} and \frac{cos(z)}{e^z+e^{-z}} with complex analysis and the residue theorem, and in the solutions they replace both sin(z) and cos(z) with e^{iz} why is this possible?

how was the trigonometric functions created? how did mathematicians find cosine, sine, tangent, etc. without a calculator. basically how would i find the trigonometric functions after the collapse of civilization and it was up to me to rewrite all the charts and program all the calculators that...

Homework Statement I'm trying to simplify some trigonometric expressions, I'm attaching my work here. This comes from a famous physics problem i.e. the rod with two masses spinning on a circle. I've tried many times but I just can't get it. Any help on which identities to use would really...

I can't seem to solve 7 cot 270° + 4 csc 90° I don't know whether I'm entering something in my calculator wrong (could've sworn I was doing it right earlier) or if there just isn't an answer. In my calculator, I enter 7/tan(270°)+4/sin(90°) and it gives me a domain error.

Prove that $\cot 7\dfrac{1}{2}^{\circ}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$.

Homework Statement lim x --> 0 for function y = (-2x)/(sinx) Using L'Hopital's Theorem, I found the derivative of the top and of the bottom and found the limit and got -2. How to find the limit as x approaches 0 without using L'Hopital's theorem. I know my solutions manual uses a...

Homework Statement It seems like a pretty straightforward equality but I when I tried to google it doesn't seem like it is known at all. All the paths I have tried have been dead ends. The question was initially: Find the limit as x approaches 0 for the expression (1-cosx)/x^2 In the...

Homework Statement I'm trying to solve ##\int\sqrt{a^2 - x^2}## by using the substitution ##x = asin\theta## Homework Equations ##x = asin\theta The Attempt at a Solution ##y = \int\sqrt{a^2 - a^2cos^2\theta}## ##y = a\int\cos\theta## ##y = a^2\int\cos(\theta)^2## ##y = (a^2)/2 *...

I'm having a problem understanding exactly why trig functions are defined the way they are. Of course, the definition in terms of 0 to 90 degree angles within right triangles is easy: the functions just give the ratio of the sides given the angle. However, I don't understand how or why trig...

Hey guys, I've a few more questions this time around from my problem set: (Ignore the first question , I only need help with 2a, b,and c.) Question: For the first one, I assessed the inside of the inverse trigonometric function first: sin^-1 (2/3) = 0.785 Then tan-1(.785) ~ .66577...

Hi to everyone. I'm new on here (in fact, this is my really first message). I need some help with the next limit, I hope you can help me: $$\lim_{x \to \infty} \sin (x\pi\sqrt [3] {x^3+3x^2+4x-5})$$ Thank you so much for your time! :)

[SIZE="4"]Definition/Summary In a right-angled triangle, with a hypotenuse ("hyp"), and with sides adjacent ("adj") and opposite ("opp") to the acute angle we are interested in, the six basic functions are defined as follows: sin = opp/hyp, cos = adj/hyp, tan = opp/adj, cosec = 1/sin, sec...

Hey guys, I'm really doubting my answer for 4b specifically. I used x=tan(Ø) and got (-1/4)cot(Ø) - (Ø) + C. I'm really not sure about this one. Thanks in advance.

Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.

my prof. gave us a bunch of homework questions and i can't seem to get around this one. integrate (x^2)/((36-x^2)^(3/2) dx (sorry. I am new to this website and i couldn't really understand how to use the commands. don't have much experience with coding. (Speechless) so i started out using...

\lim_{z \to 0} \frac{sin z}{z(z+i)} I applied L'Hopital and I got: \lim_{z \to 0} \frac{cos z}{2z+i}=\frac{1}{i} Wolphram Alpha's solution is -i. What am I doing wrong?

$0<\alpha, \beta<\dfrac {\pi}{2}$ prove :$\dfrac {1}{cos^2\alpha}+\dfrac {1}{sin^2\alpha\,sin^2\beta\, cos^2\beta}\geq 9$ and corresponding $\alpha$, and $\beta$

Evaluate $\dfrac{1}{\sin^2 \dfrac{\pi}{10}}+\dfrac{1}{\sin^2 \dfrac{3\pi}{10}}$.

Simplify $\tan x\left(1-\sec \dfrac{x}{2} \right) (1-\sec x)(1-\sec 2x)\cdots(1-\sec 2^{n-1} x)$ at $n=8$.

In certain forms - including the logarithmic - a number of the trigonometric and hyperbolic functions can be used to sum series having Riemann Zeta and Dirichlet Beta functions (in the general series term). In this tutorial, we explore some of these connections, and present a variety of Zeta and...

Find all real $k$ such that $0<k<\pi$ and $\dfrac{8}{3\sin k-\sin 3k}+3\sin^2 k\le 5$.

Solve the equation $\sin^6 a+\cos^6 a=0.25$.

Homework Statement \int\frac{\pi}{2}0 sin7y dy The bounds are from \frac{\pi}{2} to 0. Homework Equations The Attempt at a Solution I think I did the integration correctly, but I don't really know how to evaluate this. \intsiny(sin2y)3 dy \intsiny(1-cos2y)3 dy u=cosy du=-siny...

Can someone please explain to me why in this problem, for example, the sine and cosine functions are used in equating the force's components? I am having a hard time solving for these unknown forces because of my rusty trig skills. It asks to "determine the maximum weight of the flowerpot...

If $\dfrac{\sin 4x}{a}=\dfrac{\sin 3x}{b}=\dfrac{\sin 2x}{c}=\dfrac{\sin x}{d}$, show that $2d^3(2c^3-a^2)=c^4(3d-b)$.

Prove that the average of the numbers $n\sin n^{\circ}$ (where $n=2,\,4,\,6,\, \cdots,\,180$) is $\cot 1^{\circ}$.

Homework Statement Hello PF, I am taking calculus II right now, and a homework problem I came to ponder upon has been giving me big trouble today. Here is the what I have to take the integral of: ∫x/(x^(2)+8x)^(1/2) dx Every other trig substitution problems were straight forward, as all...

a) Solve the trigonometric equation: A = cosX + AsinX, for some angle X. b) The imaginary number i is equal to √-1. Use part a) to solve i = cosX + isinX. You will have used degrees to answer parts a) and b). However, the mathematics below requires the use of radians. Convert degrees to...

Given N sampled points, using the FFT we can get the Fourier transform of those N points Xk. With N/2 the Nyquist frequency and X0 the DC value. Using the inverse we can then get back the original function we just measured. However if we would like more points then just the N we have measured...

Given a trig equation, like: sin(x)² + cos(x)² = 1² or sin(x) = 1/csc(x), exist a correspondent inverse: arcsin(x) + arccos(x) = π/2 and arcsin(x) = arccsc(1/x), respectively. Thus, given an any trigonometric equation, how find its correspondent inverse?

Homework Statement A spotlight on the ground shines on a wall 14m away. A person of height 2 m walks toward the wall on a direct path between the spotlight and the wall at a rate of 5/3m/s. How fast is the height of the shadow on the wall changing when the person is 4m from the wall?Homework...

trigonometric integrals; choosing which one to "break up?" When you have two different trigonometric functions multiplied together within the integral, for example integral of (cos^4*sin^6) how do you tell which one to "break them up" to substitute an identity in? Thank you!

Show that $\displaystyle \sum_{k=0}^n \sin k=\dfrac{\sin \dfrac{n}{2} \sin\dfrac{n+1}{2}}{\sin \dfrac{1}{2}}$.

For all $x,\,y,\,z \in R$ with $x+y+z=2\pi$, prove that $\cos^2 x+\cos^2 y+\cos^2 z+2\cos x\cos y \cos z=1$

Is correct the following procediment? ## A \sin (\omega t) = A \sin (\phi) \to \phi= \sin^{-1} \sin (\omega t )## Is correct to say that ## \phi = \omega t## is oscillatory in this case ??

Here is the question: I have posted a link there to this thread so the OP can view my work.

Digging in the wiki, I found this relation between 'arc-functions' and 'arc-functions-hyperbolics" \\ arcsinh(x)= i \arcsin(-ix) \\ arccosh(x)= i \arccos(+ix) \\ arctanh(x)= i \arctan(-ix) https://it.wikipedia.org/wiki/Funzioni_iperboliche#Funzioni_iperboliche_di_argomento_complesso...

Homework Statement Composing trigonometric functions, you realize that the main substitutions are related with the table below: So, I started to integrate each expression above and I created this other table: But I had a problem with the integral circled in red, because I don't...

I'm not sure if this is the correct section for this thread since this isn't homework, but my question is very basic, so I think this section is suitable. I have two questions regarding the trigonometric functions (sinx,cosx,tanx etc). 1) What is the geometric meaning (i.e in the context...

Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$

Solve the equation $\sin a \cos b+ \sin b \cos c+ \sin c \cos a=\dfrac{3}{2}$