2
votes
3answers
26 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
1
vote
0answers
20 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
-1
votes
2answers
75 views

Scratching my head whith a problem of infinity.

The first equation in a) gives a sum of 1 and the second equation starts with a sum equal to $\pi$ or $180$ in radian and degree mode.By removing the sign from $\sqrt x$ in b) the value of y is still ...
3
votes
1answer
56 views

Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
3
votes
5answers
138 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
2
votes
1answer
53 views

Is $f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$ strictly positive?

I would like to have an advice for this exercise. Let $x\in[0,\pi]$ For which values of $x$ this function $$f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$$ is strictly positive ? I tried to ...
0
votes
3answers
44 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
3
votes
0answers
55 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
0
votes
2answers
48 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
0
votes
6answers
121 views

Why $\cos^2 x-\sin^2 x = \cos 2x\;?$

I was hoping someone could explain how $\cos^2 x-\sin^2 x = \cos 2x$ After using the product rule to differentiate $\sin x \cdot \cos x$ I get the answer $\cos^2 x - \sin ^2 x$ I've come across this ...
0
votes
1answer
33 views

Differentiate $(x-1)^2 \sin x$ where $x$ is in radians

How would I differentiate, simplify and then find $f'(\pi/2)$: $$ f(x)=(x-1)^2 \sin x $$ I'm not sure how to differentiate $\sin x$ to then use it later to find an answer, any help would be much ...
4
votes
2answers
194 views

Finding the limit of a trigonometric function

Could someone please clarify whether my calculation on the following limit problem is correct? Determine the following limit: $\lim_{x \to \frac{\pi}{2}} \frac{\sin^2x-1}{\sin x-1}$ $\lim_{x \to ...
2
votes
1answer
60 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
-2
votes
0answers
23 views

the sum of trigonometric functions and harmonic numbers [closed]

The first equation in a) gives a sum of 1 and the second equation starts with a sum equal to $\pi$. By removing $\sqrt x$ in b) the value of y is still almost the same.What is the exact value of y ...
1
vote
0answers
76 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
1
vote
2answers
50 views

What is the first step to solving $\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x)$?

My calculus BC teacher has given us some trig "review". $$\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x).$$ How do I get rewrite the cos3x and sin3x? Do I just use sum and difference, because it ...
2
votes
4answers
71 views

What is the maximum value of $ \sin x \sin {2x}$

What is the maximum value of $$ \sin x \sin {2x}$$ I have done my work here $$f (x)=\sin x \sin 2x =\frac{\cos x - \cos3x}2 $$ $$f'(x)= \frac{- \sin x+3 \sin 3x}2 =4\sin x (2-3\sin^2 x)=0$$ ...
-2
votes
3answers
84 views

$\cos 27^{\circ}~$? not using calculator .trigonometry [closed]

$\cos 27^{\circ}~$? Answer should be in square root form without using calculator. Don't use the complex form. Thanks!
7
votes
3answers
68 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.
1
vote
1answer
36 views

Why does the $\tan$ reduction formula have a restriction?

My book says the reduction formula is only valid for an integer $n > 1$. Why? This derivation doesn't require $n$ to be an integer or greater than $1$.
2
votes
1answer
19 views

Eliminating parameter to get Cartesian equation

$x = \sin(t/2)$ $y = \cos(t/2)$ $-\pi \le t \le \pi$ How would I go about getting the Cartesian equation of these?
5
votes
1answer
77 views

Show $\sin(x+h) \cdot \cos x - \cos(x+h) \cdot \sin x = \sin h$ (without limits please - straight trigonometry only).

I've tried an algebraic approach using the identity $\sin(x) = \sin(x+h-h) = \sin(x+h)\cos(h) - \cos(x+h)\sin(h)$, leading to a complicated expression I'm having trouble simplifying: ...
4
votes
4answers
100 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
2
votes
2answers
59 views

Prove that $\frac{{-\cos(x-y)-\cos(x+y)}}{-\cos(x-y)+\cos(x+y)} = \cot x \cot y$

I solved this from my implicit differentiation, and i end up with this answer, they say it's right but not simplified, I tried to simply it but I get $\cot(x)\cot(y)-\tan(x)\tan(y)$ ...
2
votes
5answers
66 views

Minimize $\cos(t)\cos(t-\alpha)$

How can I minimize $f(t)=\cos(t)\cos(t-\alpha)$? I guessed that the minimum is precisely halfway between the adjacent roots $\pi/2$ and $\pi/2+\alpha$. However, I'm not sure how to prove this. Is ...
3
votes
2answers
69 views

Derivatives of trig polynomials do not increase degree?

Let $c = \cos x$ and $s = \sin x$, and consider a trigonometric polynomial $p(x)$ in $c$ and $s$. The degree of $p(x)$ is the maximum of $n+m$ in terms $c^n s^m$. Is it the case that repeated ...
1
vote
3answers
84 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
2
votes
2answers
79 views

Integration of $1/\sin^3 x$

I need a explanation of this problem: $$ \int \frac{1}{\sin^3 x}\,dx $$ Change the variable $$ t = \tan (x/2) $$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
-1
votes
1answer
56 views

Limits of trigonometric functions as $x$ approaches to a constant $a$

$$\lim_{x \to a} \sin{x} = ?$$ $$\lim_{x \to a} \cos{x} = ?$$ What are some ways of computing these limits? I'd appreciate if you could post different methods as well.
0
votes
1answer
35 views

How to find limits involving trigonometric functions as $x\to 0$?

Problem: find the limit as $x\rightarrow 0$ of $\dfrac{\tan(3x)}{\sin(2x)}$ $\dfrac{(\sin(2x) + 3)}{(\cos(7x)-8)}$ Note I am able to solve the first one using l'Hopitals, but I really want to be ...
0
votes
2answers
58 views

Limit of $ (2^n)\sin(n) $ as $n$ goes to infinity

I'm stuck with the limit $\lim_{n\to\infty} (2^n)\sin(n) $. I've been trying the squeeze theorem but it doesn't seem to work. I can't think of a second way to tackle the problem. Any push in the right ...
3
votes
0answers
75 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
1
vote
2answers
32 views

Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$

Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$ The graph of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ appears to have a jump discontinuity at $0$ and I want to calculate the left ...
0
votes
2answers
55 views

The limit of Riemann sums $\sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$

Find the limit of Riemann sums $$\lim_{n\rightarrow \infty} \sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$$ on the interval $$[0,\frac{\pi}{2}]$$ Progress All I have managed to do is ...
6
votes
2answers
106 views

Having fun integral $\int_0^{\pi/4} \cos x \arctan(\cos x)\, dx$

Playing around with the inverse trigonometric function integration, I found a nice closed-form of the following integral $$\int_0^{\pi/4} \cos x \arctan(\cos x)\, ...
0
votes
2answers
38 views

Find the exact values without using a calculator of cos^-1(-1/2), tan^-1(-√3/3) and sec^-1 (2)

How do I solve this problems? The inverse of cosine is secant and the the inverse of tan is cotangent and the inverse of secant is cosine. Is that how I should think of it?
4
votes
6answers
146 views

How do I solve $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{4\,dx}{\sin^2(x)\cos^2(x)}$?

Alright so I have $$\int_{\pi/6}^{\pi/4}\frac{4\,dx}{\sin^2(x)\cos^2(x)}.$$ And I am not completely sure on how to tackle this problem. All I have done thus far is ...
3
votes
6answers
409 views

Trigonometric simplification for limits.

Have to evaluate this limit, but trigonometry part is :( $$\lim_{x\to 0} \dfrac{1-\cos^3 x}{x\sin2x}.$$ Had written the denominator as $2x\sin x\cos x$, no idea what to do next. Please help...
0
votes
3answers
57 views

About $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$.

$0\leq\theta<2\pi$. When $\theta$ satisfies $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$, solve $\alpha+\beta$ ( $\alpha$:= minimum $\theta$, $\beta$:= maximum $\theta$). From the graph ...
2
votes
3answers
72 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$ [duplicate]

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...
2
votes
1answer
79 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
0
votes
4answers
69 views

How can I find $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without L'Hospital's Rule

Is there a way to solve $\lim_{x \to 0}\frac{\tan(3x)}{\sin(8x)}$ without using the trig identity $\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-\tan^2(x)}$. I want to know because I had to look up this trig ...
15
votes
0answers
218 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
3
votes
3answers
60 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
0
votes
1answer
39 views

What is the graph of $y = \sin n$ and why is it different from the graph of $y = \sin x$?

I have downloaded a book about Calculus from MIT OCW. In that book, there is a section "A Thousand points of Light". (You can download the relevant section from here.) In that section, it is written ...
1
vote
9answers
142 views

Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.

I am trying to find $$\lim_{x\to0}\frac{\sin5x}{\sin4x}$$ My approach is to break up the numerator into $4x+x$. So, $$\begin{equation*} ...
2
votes
3answers
185 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
-1
votes
1answer
66 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
8
votes
2answers
191 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
0
votes
0answers
57 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.