Trigonometric Definition and 1000 Threads

Homework Statement Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө) Homework Equations I think I'm supposed to use the power reducing formulas for trigonometric identities which are sin^2(u)= (1- cos(2u))/2 cos^2(u)=(1+cos(2u))/2 *Let u represent any...

This problem is same as the problem on this link https://www.physicsforums.com/threads/trigonometric-problem.76696/ . I would like to ask the number 4 hint which is "4) We therefore have, for example the equality: " the equations can't be seen on my pc as it will only outputs this...

Dear PF Forum, Continuing our debate discussion in differential in slice of X. I read this particulare website. About proofing the derivative of sine(x). http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx In there, the web writes [FONT=Courier New] arc AC < |AB| + |BC| <...

$$\frac{\cot^3\left({y}\right)-\tan^3\left({y}\right)} {\sec^2\left({y}\right)+\cot^2\left({y}\right)} =2\cot\left({2y}\right)$$ I tried the LHS but could get it to reduce.

How can I get a right triangle from the inputs and outputs of trigonometric functions? For example: sin(x) = y The triangle would have one angle as x and the opposite edge of the triangle would be y/hyp etc. How can I get all of these values from any trigonometric function? Please tell me if I...

I am familiar with the importance of the following inverse circular/hyperbolic functions: ##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##. However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on. Given any equation of the form...

Hello. I find this whole topic of compound angle formula really confusing. I've been doing equations like cos(60+x) = sinx using the co function identities so far, yet this one seems to be impossible to do using cofunction identities so I need to know how to do it using compound angle formula...

$\int\frac{3{x}^{3}}{\sqrt{4{x}^{2}-1}}dx $ I wasn't sure what substitution to use due to what is in radical? $x=\frac{1}{2}\sec^2 \left({\theta}\right)\ dx=\frac{\sin\left({\theta}\right)}{\cos^3\left({}\right)}$

Solve sin2x= sqrt(2)/2 (using algebra) the interval is between 0 and 2pi i got the first two answers: pi/8 and 3pi/8, but i don t understand how to get the other two: 9pi/8 and 11pi/8. thanks!

I only memorized these trigonometric differential identities : `sin(x) = cos(x) `cos(x) = -sin(x) because I can convert tan(x) to sin(x) / cos(x) and sec(x) to 1 / cos(x) .. etcAnd there is no need to memorize some integral identities such as : ∫ sin(x) dx = -cos(x) + C ∫...

Homework Statement Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x## Homework Equations ##f(x) = \sin x + \cos x## ##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})## The Attempt at a Solution ##f## is increasing when ##f'(x) > 0## ##\sqrt 2 \cos(x+\frac...

Homework Statement By considering ∑z2n-1, where z=eiθ, show that Σcos(2n-1)θ=sin(2Nθ)/2sinθ. (Σ means summation from 1 to N)Homework Equations Just a guess. S=a(1-r^n)/(1-r) The Attempt at a Solution I was thinking this but it doesn't seem to work very well...

I am doig trigonometric identities and i got this one, (all will be in the picture the solution and my work) i used the double angle for this but i am afraid i didn't get the exact idea, just guessing, good guessing, so i want to know how is the proper way to reach the solution

Homework Statement [/B] ##\lim x\rightarrow \frac{\pi }{4} (\frac{1-\tan x}{\sin x - \cos x})## The Attempt at a Solution [/B] By assuming y = x-π/4 , the limit become : ## \lim y\rightarrow 0 (\frac{1- \tan (y+\frac{\pi}{4})}{\sin (y+\frac{\pi}{4}) - \cos (y+\frac{\pi}{4})}) = \lim...

Homework Statement In 2001, Windsor, Ontario received its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21 Due to the Earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can...

1) Problem: given that x is an obtuse angle for which cos^2x/(1 + 5sin^2x) = 8/35, find the value of cosx/(1 - 5 sin x) without evaluating x. 2) Relevent equations: sin(-x) = - sin x cos(-x) = cos x sin(180° - x) = sin x cos(180° - x) = - cos x sin^2x + cos^2x = 1 3) Attempt: cos^2x/(1 +...

a is the opposite side b is the adjacent side c is the hypotenuse. x is the angle Problem: expression for sin x as a function of a and c. Solution: Identities taken from textbooks: sin x = a/b (1) tan x = a/c (2) sin x = sqrt(tan^2 x / (1 + tan^2 x)) (3) substituting (2)...

Homework Statement Find the limit of : lim x-> (π/2) (2-2sin x)/(6x-3π) 2. The attempt at a solution lim x-> (π/2) (2-2sin x)/(6x-3π) =lim x-> (π/2) 2-2 sin x / 6 (x- (1/2)pi) Assuming that y = x - (π/2) So, lim y->0 (2-2sin(y+pi/2))/6y lim y->0 (2-2 (sin y cos pi/2 + cos y sin pi/2)/6y...

Homework Statement Find ## \sum_1^{23} tan^{-1}(\frac{1}{n^2+n+1}) ## Homework Equations ## tan^{-1}x + tan^{-1}y = tan^{-1}(\frac{x+y}{1-xy} )##The Attempt at a Solution I think we have to split the question in a form of relevant equation given above. First thing what should I do?

Homework Statement I have the function f(x) = cos(2x+1) I need to resolve f(x) = 0.6 when -PI <= x <= PI then put my answers in a Trigonometric Circle Homework EquationsThe Attempt at a Solution First I do this: cos(2x+1) = 0.6 and use my TI calculator to solve for X and I find the following...

Hello reader, I have an exam really soon and it includes a good bit of trigonometry, but I'm having problems with the trig stuff because this exam does not allow calculators and since I was dependent on the calculator, I haven't memorized anything about the trigonometric functions. I don't know...

Is there any way I can reduce or simplify expressions like cos(sin-1(x)), sin(cos-1(x)), cos(tan-1(x)), tan(cot-1(x)) etc.? (I refer to the arc functions, i.e. inverses, by the superscript -1, not the reciprocals)

Homework Statement Find ## \int_0^{nπ+v} |sinx| dx ## Options are, 2n+ 1+ cosv 2n+1-cosv 2n+1 2n+cosv Homework Equations Integration of sinx is -cosx.The Attempt at a Solution Sin x is + ve from 0 to π, Negative from π to 2π We can make |sinx| as sinx in 0 to π, And - sinx in π to 2π But here...

Simplify $\left(\sqrt{3}\sec \dfrac{\pi}{5}+\tan\dfrac{2\pi}{5}\right)\tan \dfrac{2\pi}{15}$.

It can be shown that the following sum: $$S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right)$$ is rational. Find the value of $S$. (Callme)

What is this asking for? Is it asking for the trig expression in the form of x and y? Or is it asking for something else?

Homework Statement Prove that: \cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)} Homework Equations I am not sure. I used factoring a sum of cubes. The Attempt at a Solution I tried \cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)} . But I...

Homework Statement Verify Lagrange's MVT for ## f(x)= sinx - sin2x ## in [ 0, π ] Homework Equations ## f'(c) = \frac{f(b)-f(a)}{b-a} ## The Attempt at a Solution Got on solving cosx= 2cos2x How to find c lies in [0, π ]? Solved it using quadratic equation but it gives a complicated value...

Homework Statement Integral of $$ x^3\sqrt{x^2+16}dx $$ answer should give $$ 1/5(x^2+16)^{5/2} -16/3(x^2+16)^{1/2}+C $$ Homework Equations x=atanθ The Attempt at a Solution Mod note: The integral is ##\int x^3 \sqrt{x^2 + 16} dx## The published answer is ##1/5(x^2+16)^{5/2}...

Hello, I am trying to solve this. This material is not covered in my class, but I still want to know how to do it. If cos(t)=$\frac{-9}{10}$ where $\pi$ <t<$\frac{3\pi}{2}$ find the values of cos(2t)= sin(2t)= cos($\frac{t}{2}$)= sin($\frac{t}{2}$)= Give exact answers, do not use decimal...

Can anyone help with this trig question, Determine the smallest angle in degrees such that sin theta+5cos theta=4 Iknow i need to use the quadratic formula but really stuck on it.

I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ... Can someone please suggest (i) an online...

Homework Statement 4sec2θ tanθ = 8 tanθ Homework Equations sec2θ = tan2θ + 1 The Attempt at a Solution 4(tan2 + 1) tanθ = 8 tanθ (4tan2 + 4) tanθ = 8 tanθ 4 tan3 + 4 tanθ = 8 tanθ 4 tan3 - 4 tanθ = 0 4tanθ(tan2θ - 1) = 0 4tanθ = 0 tanθ = 0 tan-1(tanθ) = tan-1(0) tan-1θ = 0, π tan2θ - 1 = 0...

hey there I am doing a lot of center of mass problem most of them with collisions , and most of the solutions I see, have a big part of "trigonometric trciks", like the way u watch the vectors in cm frame and the way in the lab frame. does anyone met this kind of solutions and have some tricks...

Hi, I need help proving the following trig identity, (2sinx)\overline{secxtan(2x)}=2cos^2x-csc^2x+cot^2x I have tried starting from the left hand side, the right hand side, and doing both together, but nothing seems to work. One of the ways I tried: LHS...

Hi, I need help proving the following trig identity: $$\frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)}$$ Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together...

1) Problem statement: Solve the trigonometric equation for the domain is. 0°<x<360° 5(sinx - cosx) = 4sinx - 3cosx 2) Relevent equations: secx=1/cosx cosec x=1/sinx cot x= 1/tanx tan x=sinx/cosx cot x=cosx/sinx sin^2 x + cos^2 x=1 1 + tan^2 x= sec^2 x 1+ cot^2 x = cosec^2 x Template of answer...

How prove $\cos\frac{8\pi}{35}+\cos\frac{12\pi}{35}+\cos\frac{18\pi}{35}=\frac{1}{2}\cdot\left(\cos\frac{\pi}{5}+\sqrt7\cdot\sin\frac{\pi}{5}\right)$?

(tg 570* sin 1469) / (ctg 495 * cos 781) = Its in degrees

I'm not stuck per say, but I need to know if I have the right idea for solving the rest of these questions. 1. Homework Statement For the following given trigonometric ratio and domain, determine the missing trigonometric ratio. Homework Equations cos\theta = -\frac{1}{2} ...

Mod note: Fixed the LaTeX. ##a=sinθ+sinϕ## ##b=tanθ+tanϕ## ##c=secθ+secϕ##Show that, ##8bc=a[4b^2 + (b^2-c^2)^2]## I tried to solve this for hours and have gotten no-where. Here's what I've got so far : ##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ## ## b =...

Evaluate \lim_{n \to \infty} \int_0^1 | \sin(nx)| \ dx.

Okay so I'm solving this equation: 2sin(x)-3cos(x)=1 I'm solving it by equating the left hand side to a single trigonometric equation. What I did first: I equated the left hand side to a sine function. So 2sin(x)-3cos(x)=Rsin(x+a). Then I used the addition formula. Rsin(x+a)=...

Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...

Homework Statement Calculate the complex integral along the closed path indicated: $$ \oint_C\frac{\sin{z}}{z^2+\pi^2}dz,\,\,|z-2i|=2.$$ Homework Equations $$ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} $$ $$ e^{iz}=e^{i(x+iy)}=e^{-y+ix}=e^{-y}(\cos{x}+i\sin{x}) $$ The Attempt at a Solution I really...

I read that, for ##\delta>0##, if ##\delta<z\leq\pi##, then ##\sin\frac{z}{2}\geq\frac{2\delta}{\pi}##. I cannot prove it. I know that ##\forall x\in\mathbb{R}\quad|\sin x|\leq |x|##, but that does not seem useful here... Thank you so much for any help!

I've muddled my way through the majority of my weekend assignment and I'm stuck on a problem where I need to add two formulas together. 1.) 20-10cos(x*pi/4) 2.) 30+20sin(x*pi/4) I end up with a sinusoidal function which I can then graph and determine the max, min, etc. We recently went over...

So here's the question: Suppose cos(θ) =x/4. Find expressions for the other five trigonometric functions in terms of x. In our practice problems we never had a variable x used and we were able to use the pythagorean theorem to determine the final side of the triangle and simply figure out the...

$$\lim_{{x}\to{\pi/4}} \frac{1-\tan(x)}{\sin(x)-\cos(x)}$$ So using, L'Hospital's rule, I get: $$\lim_{{x}\to{\pi/4}} \frac{\sec^2(x)}{\cos(x)+\sin(x)}$$ But $\cos(x)+\sin(x) = 0$ when $x = \dfrac{\pi}{4}$ which is an indeterminate form, so how do I go from here?

I have this problem: $\lim_{{t}\to{0}} \frac{tan(6t)}{sin2t}$ I know sin2t = 0 when t = 0, which means the original fraction is indeterminate, so how can apply the rules for limits to solve this limit?