Trigonometric Definition and 1000 Threads

Homework Statement A cubic equation is given as: ##x^{3} -(1+\cos \theta +\sin \theta)x^{2} +(\cos \theta \sin \theta +\cos \theta +\sin \theta)x-\sin \theta \cos \theta=0## Show that x=1 is a root of the equation for all values of θ and deduce that x-1 is a factor to the above equation...

Could someone help me with these two problems? I've been at them for an hour, but have very little clue how to go about solving either of them. Homework Statement 1)∫ 6 csc^3 (x) cot x dx Homework Equations The Attempt at a Solution 6 ∫ csc^3 (x) dx) / tan x csc^3 / tan x =...

Without the help of calculator, show that $\tan 50^{\circ}>1.18$

Let $k$ be an odd positive integer. Solve the equation $\cos kx=2^{k-1} \cos x$.

Homework Statement 4sin2(2x) -1 = 0 Solve over 0 --> 2pi The attempt at a solution (2sin(2x) + 1) (2sin(2x) - 1) = 0 2sin(2x) +1 = 0 sin(2x) = -1/2 2x = -pi/6 x = -pi/12 and -11pi/12 which = 23pi/12 and 13pi/12. 2sin(2x) - 1 = 0 sin(2x) = 1/2 2x = pi/6 x= pi/12 and...

Let $p,\,q,\,r,\,s\,\in[0,\,\pi]$ and we are given that $2\cos p+6 \cos q+7 \cos r+9 \cos s=0$ and $2\sin p-6 \sin q+7 \sin r-9 \sin s=0$. Prove that $3 \cos (p+s)=7\cos(q+r)$.

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

Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.

hello guys , i'm looking for approximation of trigonometric and hyperbolic functions for small and large argument, is it correct to say sin(x)=x and tg(x)=x and tgh(x)=x and cos(x) = 1 and cosh(x)=1 and coth(x)=1/x for small x what about large x ? what can we say about exponential function...

the units of all angles :degree gien :$tan\, x\,+tan\,(x+60)\,-\,tan(60-x)=3tan(3x)$ prove :$tan^2\, x\,+tan^2\,(x+60)\,+\,tan^2(60-x)=9tan^2(3x)+6$

prove: $tan 1^o+tan 5^o+tan 9^o +---------+tan 177^o=45$

$0<\alpha < \dfrac {\pi}{2}$ $0<\beta < \dfrac {\pi}{2}$ prove: $(1): \,\, \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha \, \sin^2 \beta \, \cos^2 \beta} \geq 9 $ determine the values of $ \tan \alpha$ and $ \tan \beta $ when : $(2): \: \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha...

This is one of the example problems in my book to show how to deal with integrating trigonometric functions to higher powers, by breaking them down into identities. $$=\int cos^5x dx$$ $$=\int (cos^2x)^2cos^x dx $$ $$=\int (1-sin^2x)^2*d(sin x)$$ $$=\int (1-u^2)^2 du$$ $$=\int 1-2u^2 + u^4...

$\displaystyle \int {\frac{\sqrt{x^2-9}}{x}}\ dx $ using $\displaystyle x=3\sec{\theta}\ \ \ dx=3\sin{\theta}\sec^2{\theta}\ d\theta $ so then $\displaystyle \int {\frac{3\tan{\theta}}{3\sec{\theta}}}\ 3\sin{\theta}\sec^2{\theta}\ d\theta \Rightarrow 3\int {\tan^2{\theta}}\ d\theta $ the...

Hi MHB, Solve $(2+ \sqrt{2})^{(\sin x)^2}-(2- \sqrt{2})^{(\cos x)^2}=\left( 1+ \dfrac{1}{\sqrt{2}} \right)^{\cos 2x} -(2-\sqrt{2})^{\cos 2x}$. This problem vexes me much because the only way that I could think of to solve this problem would be by substituting $(\sin x)^2=u$, and from there, I...

Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$. Solve for $y $ if $\tan 4y=\dfrac{\cos y-\sin y}{\cos y +\sin y}$.

Homework Statement Solve: \int sin(16x) \sqrt[a]{cos(16x)}\,dx Answer should be linear in the constant "a" The Attempt at a Solution \int sin(16x) \sqrt[a]{cos(16x)}\,dx Set: u=cos(16x), du=-16sin(16x) du ~~\Rightarrow~~ {-1/16}\int \sqrt[a]{u}\,du =...

Can it be proved? $$\left(\frac{-2\sin A}{1-\cos A}\right)\cos\left(\frac{A}{2}\right)\tan^{-1}\left[\cos \left(\frac{A}{2}\right)\right]=\frac{\pi^2-4A^2}{8}$$

When I place the trigonometric functions in the "wolfram google", it informs the parity of the function, so, sin(x), sinh(x) -> odd cos(x), cosh(x) -> even tan(x), tanh(x) -> odd cot(x), coth(x) -> odd sec(x), sech(x) -> even csc(x), csch(x) -> odd arcsin(x), arcsinh(x) -> odd...

Without the help of calculator, evaluate $\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\cos \dfrac{4\pi}{7}$.

how do i integrate the fifth power of a secant? i broke it up into powers of two and three but that didn't seem to work

how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)? i reduced this to sqrt(u^2-u) but i don't know how to go from here

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

Homework Statement If ##\sec x-\csc x=\pm p##, show that ##p^{2} \sin^2 2x +4\sin 2x-4=0## Show conversely that if ##p^{2} \sin^2 2x +4\sin 2x-4=0##, then ##\sec x-\csc x## is equal to +p and -p. Find, to the nearest minute, the two values of x in the range of 0 to 360 degrees, the equation...

This is my very first post - so i hope I don't break any rules - its more of a formula rearranging question/confirmation so here goes... Homework Statement Currently working on friction - static/kinetic - so in my textbook it states in a side bar "info bit" that tanθ=sinθ/cosθ my...

Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...

Stuck on this problem. Evaluate $$ \int \cos^{2}x \, \tan^{3}x \, dx$$ What I have so far: used the trig identity sin/cos = tan factored out a sin so I can have a even power. changed $$\sin^{2}x$$ to its identity = 1/2(1 - cos2x) combined like terms and canceled out the cos $$\int \cos^{2}x...

Homework Statement 4∫tan(x^2)dx from 0 to √(π)/2Homework Equations 4∫tan(x^2)dx from 0 to √(π)/2The Attempt at a Solution I tried doing u-substitution, which didn't work, and also tried to look for a trig identity and wasn't able to find any relevant one.

Quick question. $$\int sin^{4}x dx$$ so I know: $$\frac{1}{2} \int 1 - 2cos2x + \frac{1}{2}(1 + cos4x)dx$$ So here I first brought out the 1/2 because it's a constant and it's nasty. so now I have $$\frac{1}{4} \int 1 - 2cos2x + 1 + cos4x dx$$ so...Just as I brought 1/2 out can I now...

Prove that $\tan \left( \dfrac{3 \pi}{11} \right)+ 4\sin \left( \dfrac{2 \pi}{11} \right)=\sqrt{11}$. I know this problem may be stale as it has been posted countless times at other math forums, but I've seen one brilliant method to attack this problem recently, and I'm so eager to share it...

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

How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here. Here is the illustrative...

Hi MHB, I have found this problem quite interesting to me and hence I have spent some time on it but all of my attempts to prove it went down the drain. I have no choice but posting it here, hoping to gain some insight from the members of the forum on how to prove this problem. Thanks in...

Homework Statement Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt)) Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...

Homework Statement [0,1]∫(3x)dx/(4-3x)^1/2 (3xdx divided by square root of 4-3x) Homework Equations The Attempt at a Solution I could not get the bookish answer of that...actually my answer was wholly different... i let 4-3x (without square root) = t and then use substitution...

Hellow! If we can equal the first derivative with a trigonometric function: \frac{dy}{dx}=tan(\theta) So, the second derivative is equal to which trigonometric function? \frac{d^2y}{dx^2}=? Thanks!

This thread is dedicated to exploring the trigonometric series shown below. This is NOT a tutorial, so all and any contributions would be very much welcome... (Heidy)$$\mathscr{S}_{\infty}(z)= \sum_{k=1}^{\infty}\frac{\log k}{k^2}\cos(2\pi kz)$$ This series can be expressed in terms of the...

Your help will be greatly appreciated! Thanks!1. The expression \(\sin\pi\) is equal to \(0\), while the expression $\frac{1}{\csc\pi}$ is undefined. Why is $\sin\theta=\frac{1}{\csc\theta}$ still an identity? 2. Prove $\cos(\theta + \frac{\pi}{2})= -\sin\theta$

For any triangle $ABC$, prove that $\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$

Express $\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)$ in terms of $\cos 3x$.

Homework Statement please help me with this integration problem? ∫(1/sinx+ cosx) dx Homework Equations i don't know any proper substitution in this question,maybe there are none The Attempt at a Solution i tried rationalizing and it has got me this far...

$2\cos{\theta}+2\sin{\theta}=\sqrt{6}$ $\displaystyle\cos{\theta}+\sin{\theta}=\frac{ \sqrt{6} }{2}$ $\displaystyle(\cos{\theta}+\sin{\theta})^2=\frac{3}{2}$ $\displaystyle\cos^2{\theta}+2cos\theta\sin\theta+\sin{\theta}^2=\frac{3}{2}$ $\displaystyle\sin{2\theta}=\frac{1}{2} \Rightarrow...

Hi, So this is part of an assignment for my numerical analysis class. The integral is this: \int_0^{\infty} e^{-x} \cos^2 (x^2) dx We are instructed to evaluate the integral from 0 to some large A using numerical methods (which I'm fine with), and then estimate the tail, ie...

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

Homework Statement 0∫\sqrt{∏} xsin(##x^2## -1) dx Not sure how I should be formatting this, but the square root of pi is 'on top of' the integral, and zero is 'below'. The expression to integrate is \sqrt{∏} xsin(##x^2## -1) dx. The Attempt at a Solution As sin integrated is -cos...

Hello, Probably a simple problem, but I am not able to figure it out. a \cos (\epsilon) - b \sin (\epsilon) = c in-phase part a \sin (\epsilon) - b \cos (\epsilon) = d out-of-phase part In order to find the phase shift, the in-phase term has to be divided by the out-of-phase...

These are the two last problems I'll bother you with for a short while (I love this forum, I'll definitely stay on and hopefully be able to contribute in the future). Homework Statement Problem 1: (##-x^2##-1)sin2x > 0 , xe[0,2\pi] Problem 2: ##2^{-x^2+x+2}## < 4 Homework...

Homework Statement Sketch the curve of y=sin(x)+sin(2x) I got all the derivatives, and the roots N(k*∏|0), but now I'm stuck with the maxima/minima Homework Equations First derivative y'=cos(x)+2cos(2x) The Attempt at a Solution I set cos(x)+2cos(2x)=0 but now I'm stuck. I...

Working on some fluid mechanics research for flat plate boundary layer, stuck on this integral: ∫sin(∏y/2δ)dy for 0<y<δ Any help would be deeply appreciated.

Hello everyone, I've wandered PF a few times in the past but never thought I'd join, here I am, how exciting. To keep it short I'm trying to understand the proof behind Fourier Series and can't quite get to grips with basic trigonometric orthogonality. I understand that sin and cos are...