Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Prove that $\frac{dy}{dx}+\sec^2(\frac{\pi}{4}-x)=0$
If $y=\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}$, prove that $\frac{dy}{dx}+\sec^2(\frac{\pi}{4}-x)=0$
My attempts:
Attempt 1:
$y=\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}$
$\implies y=\sqrt{\frac{\sin^2 x+\cos^2 x-2\sin x \cos x}{\sin^2 x+\cos^2 x+2\sin x \cos x}}$
$\implies y=... | "Why does Attempt 2 give a result different from that given by Attempt 1."
Your mistake is that You conclude $\sqrt{a^2}=a$. Actually, one time You conclude that $\sqrt{a^2}=a$ and another time that $\sqrt{(-a)^2}=-a$ although $\sqrt{a^2}$ and $\sqrt{(-a)^2}$ are the same. Correct would be $\sqrt{a^2}=\pm a$, so two va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2813208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find out if polynomial has inverse in quotient ring Find out if polynomial $x^3-x^2+x-1$ has inverse element in quotient ring ${\displaystyle \mathbb {Z} }_{11}/(x^4+3x^3-3x^2-4x-1)$, if yes find this inverse.
I know that $x^3-x^2+x-1$ has inverse element when $gcd(x^4+3x^3-3x^2-4x-1;x^3-x^2+x-1)$~$1$. I also know that... | Since Jose already covered the fact that your approach is valid, I'll present a cool alternative approach. First, construct the companion matrix of $x^4+3x^3-3x^2-4x-1$, call it $A$.
Here $$A = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -3
\end{pmatrix}_.$$
Now consider the matrix, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2815784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Simplify: $\frac{1+\cos2x}{1-\cos x}-\frac{4\cos^2(\frac{x}{2})}{\tan^2x}$ Simplify:
$$\frac{1+\cos2x}{1-\cos x}-\frac{4\cos^2(\frac{x}{2})}{\tan^2x}$$
So what I've tried was:
$$\frac{1+\cos2x}{1-\cos x}-\frac{4\cos^2(\frac{x}{2})}{\tan^2x}=\frac{1+\cos^2x-\sin^2x}{1-\cos x}-\frac{4\cos^2(\frac{x}{2})}{\tan^2x}=$$
$$=\... | We have that
$$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1$$
$$\cos^2\left(\frac{x}{2}\right)=\frac12(\cos x +1)$$
$$\tan^2x=\frac{1-\cos^2x}{\cos^2 x}$$
thus
$$\frac{1+\cos2x}{1-\cos x}-\frac{4\cos^2(\frac{x}{2})}{\tan^2x}=$$
$$\frac{2\cos^2x}{1-\cos x}-\frac{2(\cos x +1)}{\frac{1-\cos^2x}{\cos^2 x}}=$$
$$\frac{2\cos^2x}{1-\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2818753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Bitstrings and set of solutions There is something about counting bitstrings and the format of the solutions I didn't really understand yet.
Given a bitstring problem that asks to elementarily count how many bitstrings of length 36 there are that contain exactly 5 $\{1\}$ in the first 10 posititions and the $\{10100101... | In order to calculate the number of binary strings of length $20$ containing the substring $10100101$ pretty much of all the hard work is already done. We just have to use the inclusion-exclusion principle and put all the intermediate results together.
The number of wanted strings is
\begin{align*}
&\binom{13}{1}2^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2820787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability that all three shirts were worn Mr. Jones has three shirts: Red, Green and White. Each day he picks randomly a red shirt with probability of $\frac{1}{2}$, green with probability of $\frac{1}{3}$ and white with probability of $\frac{1}{6}$.
What is the probability that he wears all three shirts after 6 day... | Time to use inclusion-exclusion: the probability that he'll use all three is the probability that the sequence cannot come from a corpus of just a given two.
So we have: $$P(\text {all of }RGW)=P(RGW)-P(RG)-P(RW)-P(GW)+P(R)+P(G)+P(W)-P(\emptyset)$$. Which is relatively simple: each $P$ is the summed probability of get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2821454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Finding $C_n=4C_{n-1}+6n-1$ 2.$$\begin{cases}
C_n=4C_{n-1}+6n-1\\
C_0=2\\
\end{cases} $$
$\begin{cases}
C_n=D_n+dn+e\\
D_n=\lambda D_{n-1}\end{cases}$
$D_n+dn+e=4C_{n-1}+6n-1$
But $C_{n-1}=D_{n-1}+d(n-1)+e$
So $D_n+dn+e=4(D_{n-1}+d(n-1)+e)+6n-1$
$D_n+dn+e=4D_{n-1}+4dn-4d+4e+6n-1$
But for $\lambda=4$ we get $4D_{n-1}+... | $$C_1=4C_0+6\cdot1-1,$$
$$\frac{1}{4}C_2=C_1+\frac{1}{4}(6\cdot2-1),$$
$$\frac{1}{4^2}C_3=\frac{1}{4}C_2+\frac{1}{4^2}(6\cdot3-1),$$
$$.$$
$$.$$
$$.$$
$$\frac{1}{4^{n-1}}C_n=\frac{1}{4^{n-2}}C_2+\frac{1}{4^{n-1}}(6n-1),$$
which after summing gives
$$\frac{1}{4^{n-1}}C_n=4\cdot2+6\left(1+\frac{2}{4}+\frac{3}{4^2}+...+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2823765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How do I solve this system of non-linear differential equations? How do I solve this system of differential equations?
\begin{equation}
\frac{d^2x}{dt^2}+2\frac{1}{y}\frac{dx}{dt}\frac{dy}{dt}=0 \quad\quad
\frac{d^2y}{dt^2}+\frac{1}{y}
\left[\left(\frac{dx}{dt}\right)^2-\left(\frac{dy}{dt}\right)^2\right]=0
\end{equat... | Rearranging the first equation and integrate
\begin{align} \frac{\ddot x}{\dot x} + \frac{2\dot y}{y} &= 0 \\
\ln(\dot x) + 2\ln(y) &= \ln(a)
\end{align}
$$ \implies \dot x = \frac{a}{y^2} $$
Plugging this into the second equation
$$ \ddot y + \frac{1}{y}\left[\frac{a^2}{y^4} - (\dot y)^2\right] = 0 $$
This equation i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2827823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral Related to Exponential Function and Modified Bessel Function 1 I'm looking for a closed form solution for the following integral:
$$S(x) = \displaystyle \int\limits_0^x r \,I_0(r) \,e^{-b^2r^2} \,dr $$
where $I_0()$ is the $0$-th order modified Bessel function of first kind, $b>0$, and $x>0$.
Thanks.
| $\int_0^xrI_0(r)e^{-b^2r^2}~dr$
$=\int_0^x\sum\limits_{n=0}^\infty\dfrac{r^{2n+1}e^{-b^2r^2}}{4^n(n!)^2}~dr$
$=\int_0^x\sum\limits_{n=0}^\infty\dfrac{r^{2n}e^{-b^2r^2}}{2^{2n+1}(n!)^2}~d(r^2)$
$=\int_0^{x^2}\sum\limits_{n=0}^\infty\dfrac{t^ne^{-b^2t}}{2^{2n+1}(n!)^2}~dt$
$=-\left[\sum\limits_{n=0}^\infty\sum\limits_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2828319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
integration by parts $\int \frac{x^{2}+4x}{x+2}\,dx$ $$\int \frac{x^{2}+4x}{x+2}\,dx$$
I have written it in this form:
$$\int \frac{(x+2)^{2}-4}{x+2}\,dx$$: on this stage I try to do integration by parts, which gets me to :
$$\frac{x^{2}}{2}+2x-4\ln\left | {x+2} \right |$$
but it's wrong for some reason. The right answ... | Also, it should be $$\frac{x^{2}+4x+4}{2}-4\ln\left | x+2 \right |+C_1=\frac{x^{2}+4x+4}{2}-4\ln(x+2)+C_1$$ for $x>-2$ and
$$\frac{x^{2}+4x+4}{2}-4\ln\left | x+2 \right |+C_2=\frac{x^{2}+4x+4}{2}-4\ln(-x-2)+C_2$$ for $x<-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2828775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Proving the product of four consecutive integers, plus one, is a square I need some help with a Proof:
Let $m\in\mathbb{Z}$. Prove that if $m$ is the product of four consecutive integers, then $m+1$ is a perfect square.
I tried a direct proof where I said:
Assume $m$ is the product of four consecutive integers.
If $m... | Write the product of the four consecutive integers starting at some $n-1$, so that
$$m=(n-1)n(n+1)(n+2)+1,$$
and expand:
\begin{align}
m&=(n^2-1)(n^2+2n)+1=n^2(n^2-1)+2n(n^2-1)+1 \\
&=(n^2-1)^2+2n(n^2-1)+\not 1+ n^2{-}\!\not1 \\
&= \bigl((n^2-1)+n\bigr)^2.
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2832986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 13,
"answer_id": 8
} |
Showing that $\sin(\sqrt{4 \pi^{2}n^{2} + x})$ converges uniformly on $[0,1]$
Suppose we are considering the sequence of functions $f_{n}(x)=\sin(\sqrt{4 \pi^{2}n^{2} + x})$ and I am having trouble showing that that $f_{n}$ converges uniformly on the interval $[0,1]$.
An idea, I've tried is to consider the Taylor s... | Fix $x\in[0,1]$. The Mean Value Theorem shows that $$|\sqrt{4\pi^2n^2+x}-2\pi n|<\frac x{4\pi n}\le\frac1{4\pi n}.$$ So $2\pi n \le \sqrt{4\pi^2n^2+x} < 2\pi n +1/(4\pi n)$. Then apply MVT again to show$$|\sin(\sqrt{4\pi^2n^2+x})|
=|\sin(\sqrt{4\pi^2n^2+x})-\sin(2\pi n)|<\frac1{4\pi n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2833814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Evaluate: $\lim_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$ without L'Hopitals I have to find the limit of $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$
here is my try $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}=\lim\limits_{x \to \pi/4}\frac{4(1-\c... | Observe that
$$
\cos x+\sin x=\sqrt{2}\cos(x-\pi/4)
$$
so with the substitution $x=t+\pi/4$ and noting that $\sin2x=\sin(2t+\pi/2)=\cos2t$, the limit becomes
$$
\lim_{t\to0}\frac{8-8\cos^5 t}{1-\cos2t}=
8\lim_{t\to0}\frac{1-\cos t}{1-\cos 2t}(1+\cos t+\cos^2t+\cos^3t+\cos^4t)
$$
that's easy to manage.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2835175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Final all $x$ such that $|\sqrt{x} - x| < 1$ (simple but can’t resolve) I’m a beginner so be kind with me, I tried to resolve this for 2 hours now but always give me the same wrong result. I think I’m missing something big...
Can someone help me with this?
I began by assuming $x\geq 0$ (root) and then I started to ma... | For $x\geq 0$ you have that
$|x-\sqrt{x}|=|(\sqrt{x}-\frac{1}{2})^2-\frac{1}{4}|<1$ if and only if
$-\frac{3}{4}<(\sqrt{x}-\frac{1}{2})^2<1+\frac{1}{4}=\frac{5}{4}$
So
$-\frac{\sqrt{5}}{2} <\sqrt{x}-\frac{1}{2}<\frac{\sqrt{5}}{2}$
and
$\frac{1}{2}-\frac{\sqrt{5}}{2} <\sqrt{x}<\frac{1}{2}+\frac{\sqrt{5}}{2}$
$0\leq x<... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2835637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Solve the equation $X^2+X=\text{a given matrix}$ I want to solve the quadratic matrix equation
$$X^2+X=\begin{pmatrix}1&1\\1&1\end{pmatrix}$$
If I put $X$ in the form
$$X=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$
then I find complicated equations. Is there a simple way to tackle the problem without using diagonalization... | The spectral mapping theorem gives $$\{\lambda^2 +\lambda : \lambda \in \sigma(X)\} = \sigma(X^2 + X) = \sigma\begin{pmatrix}1&1\\1&1\end{pmatrix} = \{0,2\}$$
It follows that we have one of the four options:
*
*$\sigma(X) = \{0,1\}$ so $\operatorname{Tr} X = 1$.
A calculation with $X = \pmatrix{a & b \\ c & 1-a}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2836028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 1
} |
Finding the recursive formula for calculating the solution of the linear equation system $A$x = b. Let $K$ be a field and n $\in$ $\mathbb{N}$. It's given that $A$ = $
\begin{bmatrix}
0 & p_{1} & 0 & 0 & \dots & 0 \\
q_{2} & 0 & p_{2} & 0 & \dots & 0 \\
0 & q_{3} & 0 & p_{3} & \dots & 0 \\
\vdots & & \ddots & \ddots... | Here are $A^{-1}$ for $n$ from $1$ to $4$:
$$(1/p_1),\ \pmatrix{ -p_2/(p_1 q_1) & 1/q_1\cr 1/p_1 & 0\cr},\
\pmatrix{p_2 q_2/(p_1 q_1 p_3) & 1/q_1 & -p_2/(q_1 p_3) \cr 1/p_1 & 0 & 0\cr -q_2/(p_1 p_3) & 0 & 1/p_3},\
\pmatrix{-p_2 q_2 p_4/(p_1 q_1 p_3 q_3) & 1/q_1 & p_2 p_4/(q_1 p_3 q_3) & -p_2/(q_1 q_3)\cr
1/p_1 & 0 & 0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2837975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$(\dim_K\ker f)^2 + (\dim_K\operatorname{im} f)^2 = (\dim_K X )^2$ So I have $X$, $Y$ vector spaces over a field $K$ and $f : X\to Y$ linear operator.
The problem asks me to define $\ker f$. Next, it says that given
$(\dim_K\ker f)^2 + (\dim_K\operatorname{im} f)^2 = (\dim_K X)^2$ and $\dim_K\ker f \neq 0$
I must show... | By the Rank-Nullility Theorem:
$$\dim_K \ker f + \dim_K \operatorname{im} f = \dim_K X\tag{1}$$
Now
$$(\dim_K\ker f)^2 + (\dim_K\operatorname{im} f)^2 = (\dim_K X)^2\tag{2}$$
and $\dim_K\ker f \neq 0$. Leting $\dim_K \ker f=a$, $\dim_K \operatorname{im} f =b$, and $\dim_K X=c$, for some $a$, $b$, $c\in\Bbb{Z}_{\ge0}$t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2839882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$
Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$
It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that?
I've tried to write them either:
$$5^x+7^x+11^x=2^x*3^x+2^{3x}+3^{2x}$$
or
$$... | Another elemenatary solution, using that $x$ is supposed to be a natural number. It even uses a technique the OP considered:
The equation is equivalent to
$$\left(\frac{5}{11}\right)^x + \left(\frac{7}{11}\right)^x +\left(\frac{11}{11}\right)^x = \left(\frac{6}{11}\right)^x +\left(\frac{8}{11}\right)^x +\left(\frac{9}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
When is $n^2-1$ a sum of two squares? I am trying to work out when $n^2-1$ is a sum of two squares. Is there a formula for such $n$? I have found $n=1$, $n=3$ and $n=9$ so far but am struggling to find a pattern that will generalise. If there is no such pattern then how might I prove that there are infinitely many such... | Here is a proof that there are infinitely many such $n$. Note that for any $m\in\mathbb{N}$, both $m^2=m^2+0^2$ and $m^2+1=m^2+1^2$ are a sum of two squares. It follows that both $2m^2$ and $2m^2+2$ are a sum of two squares (since a product of two sums of two squares is a sum of two squares). Letting $n=2m^2+1$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2841530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$? I know that I have to use the AM-GM inequality.
I tried separating the fraction:
$$\frac{4x^2+2x+7}{6(x+1)} + \frac{6(x+1)}{6(x+1)}$$
However, it doesn't seem to make either side of the inequality into a number.
I would appreciate some help, tha... | Hint: by polynomial euclidean division $\;\dfrac{4x^2+8x+13}{6(x+1)} = \dfrac{1}{6}\left(4(x+1) + \dfrac{9}{x+1}\right) \ge \ldots\,$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
If $a_n\sim b_n$ decrease to $0$ and $a_{n+1}b_n-a_nb_{n+1}$ changes sign i.o. , does it imply the same for $(1-a_{n+1})(1-b_n)-(1-a_n)(1-b_{n+1})$?
Suppose $a_n<b_n$ for all $n$, both are strictly decreasing to $0$ and $a_n\sim b_n$. If $A_n=a_{n+1}b_n-a_nb_{n+1}$ changes sign infinitely often, does it follow that $B... | $\def\peq{\mathrel{\phantom{=}}{}}$Take an arbitrary $c > 1$ and define$$
a_n = \frac{1}{c^n + c^{\frac{3}{4} \left[ \frac{n}{2} \right] + \frac{n}{2}}}, \quad b_n = \frac{1}{c^n}. \quad \forall n \geqslant 1
$$
It is easy to verify that $a_n < b_n$, $a_n \sim b_n\ (n → ∞)$, and both $\{a_n\}$ and $\{b_n\}$ strictly de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Geometric or matrix intuition on $A(A + B)^{-1}B = B (A + B)^{-1} A$ I am curious about a seemingly simple identity in matrix algebra. Though matrix multiplication is not commutative (the classic example of noncommutativity, it does allow a commutativity of sorts around a very specific third matrix:
$$
\color{blue}{A ... | Just put
$$
A = \left( {A + B} \right) - B
$$
to get
$$
\eqalign{
& \left( {\left( {A + B} \right) - B} \right)\left( {A + B} \right)^{\, - \,1} B = B\left( {A + B} \right)^{\, - \,1} \left( {\left( {A + B} \right) - B} \right) \cr
& \quad \Downarrow \cr
& B - B\left( {A + B} \right)^{\, - \,1} B = B - B\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Is there a way to know if a row reduction of a matrix has been done correctly? I'm an undergrad taking the class of "Linear algebra 1". I came across a problem:
sometimes we need to apply Gaussian elimination for matrices. Very quickly this skill is not much necessary as it's not a thinking skill but purely Technic.
Y... | The actual algorithm for Gaussian Elimination looks like this.
I answered a similar answer showing how to perform the LU decomposition which is Gaussian Elimination without pivoting The purpose is to zero out the row beneath is with $ \ell_{jk}$. That is why you have the operation below $ u_{j,k:m} = u_{j,k:m} - \el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Is there a way to simplify $\prod_{i=1}^n\cos(a^i\theta)$, where $a<1$? I recently came upon the following expression in an attempt at getting a closed-form solution for a recursive relation:
$$\prod_{i=1}^n \cos(a^i\theta)$$
where $a<1$. Is there a way to make this product into a sum or otherwise make it simpler, or a... | Note this is @saulspatz answer with the computations added (I couldn't resist).
With
$$
f(\theta)=\prod_{k=1}^n\cos(a^k\theta)
$$
Then noting
$$
-\int_0^{a^k\theta}\tan x\, \mathrm{dx}=\log\cos(a^k\theta)
$$
we have on taking the logarithm
\begin{align*}
\log f(\theta) &= \sum_{k=1}^n\log\cos(a^k\theta)=\log\cos(a\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove there is no $x, y \in \mathbb Z^+ \text{ satisfying } \frac{x}{y} +\frac{y+1}{x}=4$ Prove that there is no $x, y \in \mathbb Z^+$ satisfying
$$\frac{x}{y} +\frac{y+1}{x}=4$$
I solved it as follows but I seek better or quicker way:
$\text{ Assume }x, y \in \mathbb Z^+\\ 1+\frac{y+1}{y}+\frac{x}{y} +\frac{y+1}{x... | For $x,y\in \Bbb Z^+$ we have $$\frac {x}{y}+\frac {y+1}{x}=4\implies x^2-4xy+y^2+y=0\implies x=2y\pm \sqrt {3y^2-y}\implies$$ $$\implies \exists z\in \Bbb Z^+\;( z^2=3y^2-y=y(3y-1))\implies$$ $$ \implies\exists a,b \in \Bbb Z^+\;( y=a^2 \land 3y-1=b^2)\implies$$ $$\implies\exists a,b\in \Bbb Z^+\;(3a^2-1=b^2)\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Angles on a point inside a triangle Let $ABC$ be an isosceles triangle with $AB=AC$ and $∠BAC = 100$. A point $P$
inside the triangle $ABC$ satisfies that $∠CBP=35$ and $∠PCB= 30$. Find the
measure, in degrees, of angle $∠BAP$. Attached is the figure of the triangle
I tried to Angle Chase but it seemed true for all ... | Let $x=\angle APB$, $a=AB=AB$, $b=AP$. We have
$$\frac{\sin x}{a}=\frac{\sin 5^\circ }{b}$$
and
$$\frac{\sin \left(245^{\circ} - x\right)}{a}=\frac{\sin 10^\circ }{b}$$
We have then:
$$\frac{\sin \left(245^{\circ} - x\right)}{\sin x}=\frac{\sin 10^\circ }{\sin 5^\circ }$$
Using the formula
$$\sin(\alpha-\beta)=\sin\al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.
If $a,b,c,d > 0$ and distinct then show that
$$
\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c}
\ge \frac{16}{3(a+b+c+d)}
$$
I tried using HM < AM inequality but am missing on $16$. Probably... | Hi I don't think this is true. For example if $a=b=c=d=1$ then we have $4/3 >= 4$ which is false.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$ If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$ (equal sets), where $a$, $b$, and $c$ are positive integers.
This is math contest problem (I don't know the source). I was struggling to solve t... | HINT.-For some positive integers $m,n$ we have
$$\begin{cases}a+b=m(c^2+1)\\abn=c(c^2-c+1)\end{cases}\Rightarrow c^2-\frac{a+b}{m}c+abn=0$$
If $m$ and $n$ are equal to $1$ then $c$ is root of the equation $X^2-(a+b)X+ab=0$ from which clearly $c\in\{a,b\}$. It easily follows that $a$ or $b$ is equal to $c^2-c+1$.
If bot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2860229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Adding always the half of the previous number to a number beginning with 1... This question blows my mind.
Let's say we start at 1 and keep adding always 0.5 * the previous number.
1 + 0.5 + 0.25 + 0.125 + .....
The question is: will it reach infinity?
I really don't know, kept thinking about this alot.
| Consider calculating the sum in steps, at each step adding the next term. At the $n^{th}$ step, you're adding $1/2^n$ (adding $1$ is the $0^{th}$ step). Then at the $n^{th}$ step, the total sum is the following:
$$
s_n = 1 + \frac{1}{2} + \frac{1}{4} + ... + \frac{1}{2^n}
$$
Now remark that you can perform the followin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Finding the minimum value of $\log _d a + \log _bd + \log _ac + \log _c b$ The question says to find the minimum value of $$\log _da + \log _bd + \log _ac + \log _cb$$ Given, $a,b,c,d \; \in R^+ -\Bigl(1\Bigl)$
My approach:
I used the AM-GM inequality and so we can see that $$\frac{\log _da + \log _bd + \log _ac + \log... | Hint.
Assuming positive values, You can use also
$$
\log_y x = \frac{\log x}{\log y}
$$
then
$$
\frac{\log a}{\log d}+\frac{\log d}{\log b}+\frac{\log c}{\log a}+\frac{\log b}{\log c}\ge 4
$$
NOTE
Another approach.
Assuming $a\ne 0,b\ne 0,c\ne 0,d\ne 0\;\;$ and calling
$$
f(a,b,c,d) = \frac ad+\frac db+\frac ca+\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Between any two powers of $5$ there are either two or three powers of $2$ Is this statement true?
Between any two consecutive powers of $5$, there are either two or
three powers of $2$.
I can see that this statement is true for cases like
$$5^1 < 2^3 < 2^4 < 5^2$$
or
$$5^3 < 2^7 < 2^8 < 2^9 < 5^4$$
But I am havi... | Basically, it boils down to the fact that $5$ is between $2^2 = 4$ and $2^3 = 8$. Here's a proof, though.
Let $5^a$ and $5^{a+1}$ be the two consecutive powers of $5$. Let $2^b$ be the smallest power of $2$ that exceeds $5^a,$ and $2^c$ the largest below $5^{a+1}$.
Then we have $2^{b-1} \leq 5^a < 2^b$ and $2^c < 5^{a+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 4
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$|a^{2x}+a^{x+2}-1|\ge 1$ equation for positive a If $|a^{2x}+a^{x+2}-1|\ge 1$ for all values of a(a>0), $\ne 1$. Find the domain of x.
I tried to substitute $a^x=t$ and used the following $|
t^{2}+a^2t-1|\ge 1$ but it is getting complicated.
| Assuming that $a, t > 0$,
If $t^2 + a^2 t - 1 < 0$, then
\begin{align}
|t^2 + a^2 t - 1| &\ge 1 \\
1 - a^2 t - t^2 &\ge 1 \\
t^2 + a^2 t &< 0
\end{align}
Which has no positive real solution.
If $t^2 + a^2 t - 1 \ge 0$, then
\begin{align}
|t^2 + a^2 t - 1| &\ge 1 \\
t^2 + a^2 t - 1 &\ge 1 \\
(t + \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integration of $\frac{1}{x^2-a^2}$ by trigonometric substitution?
$$\int \frac{1}{x^2-a^2}dx$$
Now, I know this can be done by splitting the function into two integrable functions,
$\displaystyle\dfrac{1}{2a}\int \bigg(\dfrac{1}{x-a} - \dfrac{1}{x+a}\bigg)dx$
And then doing the usual stuff.
My question is, how can we... | We have
$$I=\int \frac{1}{x^2-a^2}dx$$
Let $x=a\cos\theta$, then $dx=-a\sin\theta d\theta$.
\begin{align}
I
&= \int \frac{1}{x^2-a^2}dx \\
&= \int \frac{-a\sin\theta}{a^2(\cos^2\theta-1)}d\theta \\
&=\int \frac{-a\sin\theta}{-a^2\sin^2\theta}d\theta \\
&= \frac1a\int \csc\theta d\theta \\
&= -\frac1a\ln\left\lvert\csc\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2866731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find the least value for $\sin x - \cos^2 x -1$
Find all the values of $x$ for which the function $y = \sin x - \cos^2 x -1$ assumes the least value. What is that value?
At first I found the first derivative to be $y' = \cos x + 2 \sin x \cos x$.
Critical point $0$, $-π/6$ (principal)
$y'' = - \sin x + 2(\cos 2x)$
T... | The critical points in $[-\pi,\pi]$ are those for which
$$
\cos x(1+2\sin x)=0
$$
which means $x=\pi/2$, $x=-\pi/2$, $x=-\pi/6$ and $x=-5\pi/6$.
Note that
$$
y''=-\sin x+2\cos^2x-2\sin^2x
$$
Since
$$
y''(\pi/2)=-1-2=-3,
\quad
y''(-\pi/2)=1-2=-1,
\\
y''(-\pi/6)=\frac{1}{2}+2\frac{3}{4}-2\frac{1}{4}=\frac{3}{2}=y''(-5\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Showing that $\left|\frac{1}{z^4+1}\right|\leq\frac{1}{1-r^4}$
I am trying to show that if $|z|=r<1$, then
$$\left|\frac{1}{z^4+1}\right|\leq\frac{1}{1-r^4} \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
I have shown the inequality $$\left|\frac{1}{z^3+1}\right|\leq\frac{1}{1-r^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ holds under t... | By the reverse triangle inequality it holds that $|z^4+1| = |1-(-z^4)| \geq 1-|z|^4$ and therefore $\frac{1}{|z^4+1|} \leq \frac{1}{1-|z|^4} = \frac{1}{1-r^4}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2870848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
[Integral][Please identify problem] $\displaystyle\int \cfrac{1}{1+x^4}\>\mathrm{d} x$ Here is my attempt. The result is not right. Please help identify the issue(s).
$\displaystyle f(x)=\int\cfrac{1}{x^4+1}\>\mathrm{d}x$, let $x=\tan t$, we have $ \mathrm{d}x = \sec^2 t\>\mathrm{d}t,\> t=\tan^{-1} x\in\left(-\cfrac{\p... | First of all, you made a typo in the final answer — the correct answer must be
$$f(x)=\frac{\sqrt{2}}{4}\tan^{-1}\frac{\sqrt{2}x}{1\color{red}{-}x^2}+\frac{\sqrt{2}}{8}\ln\frac{\sqrt{2}(x^2+1)+x}{\sqrt{2}(x^2+1)-x}+C.$$
The next issue is the introduction of $\sec(2t)$ and $\tan(2t)$ when you switched to
$$\int\frac{\se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2873537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Solve the equation $\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$ Solve the equation:
$$\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$$
Things I have done so far:
$$\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$$ Deducting 2 from both sides of equation
$$\Leftrightarrow (\sqrt[3]{x^2+4}-2)=(\sqrt{x-1}-1)+2x-4$$
$$\Leftrightarrow \frac{x^2+4-8}{\sqrt[3]{... | $$\sqrt[3]{x^2+4}=\sqrt{x-1}+2x-3$$
Let $x=t^2+1$ with $t \ge 0$. Then
\begin{align}
\sqrt[3]{t^4+2t^2+5}-t &= 2t^2-1 \\
\sqrt[3]{t^4+2t^2+5} &= 2t^2+t-1 \\
t^4 + 2t^2 + 5 &= 8t^6 + 12t^5 - 6t^4 - 11t^3 + 3t^2 + 3t - 1 \\
8t^6 + 12t^5 - 7t^4 - 11t^3 + t^2 + 3t - 6 &= 0 \\
(8t^5 + 20t^4 + 13t^3 + 2t^2 + 3t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2875620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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$x+y+z=n$. Finding the number of solutions. I have found two formulas. I want to connect them!
The number of ways in which a given positive integer $n≥3$ can be expressed as a sum of three positive integers $x,y,z$ (i.e. $x+y+z=n$) , subject to $x≤y≤z$ is the integer closest to $\frac{n^2}{12}$.
If $x+y+z=n$ where $x,... | Just by looking at the generating functions, we can see a link (colors applied) ...
The number of solutions for $x+y+z=n, x\geq1, y\geq1, z\geq1$ is the coefficient of $x^n$ term of the
$$(x+x^2+x^3+...+x^k+...)^3=\color{red}{\frac{x^3}{(1-x)^3}}$$
But because
$$\color{blue}{\frac{1}{(1-x)^3}=
\frac{1}{2}\left(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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System of non-lineair equations I can't find the solutions of this system of equations. I only need the "whole number" solutions. The solutions I'm searching for:
$x=-1 \land y = 1 \land \lambda = \frac{1}{2e}$
\begin{cases} \begin{array} {rcl} e^{x+y-y^2}+2\lambda x \ = 0 \\ e^{x+y-y^2}(1-2y)+2\lambda y\ = 0 \\ x^2+... | Given $(x,y,z)\in \mathbb{Z}$, $x^2+y^2-2=0$ is satified by $x = \pm 1,y = \pm 1$. Then
\begin{eqnarray*}
e^{x+y-y^{2}}+2\lambda x &=& 0
\end{eqnarray*}
can have the following possibilities,$e+2\lambda=0$, $\frac{1}{e}-2\lambda=0$, $\frac{1}{e}+2\lambda=0$ or $\frac{1}{e^{3}}-2\lambda=0$. This lead to $\lambda=\left\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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If $p_A(x) = x^4 (x+3)^2 (x-4)$ then $A$ is diagonalizable iff $\operatorname{Rank}(A) + \operatorname{Rank}(-3I-A) = 8$
Given the Characteristic polynomial of a matrix $A$ is
$$
p(x) = x^4 (x+3)^2 (x-4),
$$
show that $A$ is diagonalizable if and only if
$$
\operatorname{Rank}(A) + \operatorname{Rank}(-3I-A)... | Look at the possible Jordan form of $A$
$$
\begin{bmatrix}
-3 & * & 0 & 0 & 0 & 0 & 0\\
0 & -3 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 4 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & * & 0 & 0\\
0 & 0 & 0 & 0 & 0 & * & 0\\
0 & 0 & 0 & 0 & 0 & 0 & *\\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
$$
(* elements are $0$ or $1$) to deduce that
*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2880037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to proof the reason? I have this statement:
If $\frac{a}{b} = \frac{c}{d},$ prove that
$\frac{a+b}{a-b}=\frac{c+d}{c-d}$
I tried to add 1, multiply 1 and nothing.
My development was:
$\frac{a}{b} - \frac{b}{b} = \frac{c}{d} - \frac{d}{d}$
$\frac{a-b}{b} = \frac{c-d}{d}$
$\frac{b}{a-b} = \frac{d}{c-d}$ (I raised... | Actually the result is not true if $a = b$ and $\frac ab =\frac cd = 1$ and $c = d$. Then $\frac {a+b}{a-b}$ and $\frac {c+d}{c-d} $ are both undefined and not equal to each other.
.......
Three ideas:
1) Proportions:
$\frac ab = \frac cd$ means there is some ratio so that $a = cr$ and $b = dr$. (That's clear, isn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x,y,z>0$, $x+y+z=1$. Prove $x^2+y^2+z^2+3xyz \geq \frac{4}{9}$
Let $x$, $y$, $z$ be positive real numbers such that $x+y+z=1$, then
$$ x^2+y^2+z^2+3xyz \geq \frac{4}{9}.$$
From Cauchy–Schwarz inequality, we have
$$ \left( x^2+y^2+z^2 \right)\left(1^2+1^2+1^2\right)\geq(x+y+z)^2 = 1 \implies x^2+y^2+z^2 \geq \frac{... | Rewrite your last line as:
$$2\sum_{x,y,z}(x^3+y^3-xy^2-x^2y) = 2\sum_{x,y,z}(x-y)^2(x+y)\geq 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2884284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Proving divisibility by $3$ For all integers $a$, there exists an integer $b$ so that $3 | a + b$ and $3 | 2a + b$.
So far I have been able to find an integer $b$ that satisfies both of them separately, but not at the same time. (For the first one I have $b = 6-a$, and for the second I've found $12-2a$)
| Divisibilty by 3 can be easily be proved by modular arithmetic.
Suppose we have a number $A$ which is :
$A = a_n \cdot 10^n+a_{n-1} \cdot 10^{n-1}+a_{n-2} \cdot 10^{n-2}+ \cdots + a_2 \cdot 10^2 +a_1 \cdot 10+ a_0 $
We know
$1 \equiv 10 \pmod{3}$
Therefore we substitute 10 s with 1 s :
$A = a_n \cdot 10^n+a_{n-1} \cdo... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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If the equation $2x+2y+z=n$ has $28$ solutions, find the possible values of $n$
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions, find the possible values of $n$.
I tried taking $2x$ as $a$, and then $2y$ as $b$, and then finding the possibilities. However, I am not sure about the approac... | I assume that the equation $2x+2y+z=n$ has $28$ solutions when $x,y,z$ are positive integers.
Note that this is same as finding the powers of $t$ with coefficient $28$ in the following expansion.
$$(t^2+t^4+t^6+t^8+.......)^2(t+t^2+t^3+t^4+......)$$
$$\left(\dfrac{t^2}{1-t^2}\right)^2\times\dfrac{t}{1-t}=t^5\times\left... | {
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Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$? Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$ ?
For $a$ and $b$ are both positive or both negative,I proved this.
But I am not able to prove for ... | The first inequality is equivalent to $a^2+b^2\leq 2( a^2+b^2)+2ab$, i.e., to $0\leq (a+b)^2$, which obviously is true.
The second is equivalent to $2( a^2+b^2+ab)\leq 3( a^2+b^2)$, i.e., to $0\leq (a-b)^2$ which again is obviously true.
| {
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"source": "stackexchange",
"question_score": "1",
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Prove that $\frac{1+2\sin{\theta}\cos{\theta}}{\cos^2{\theta}-\sin^2{\theta}}=\frac{1+\tan{\theta}}{1-\tan{\theta}}$ Prove that $$\frac{1+2\sin{\theta}\cos{\theta}}{\cos^2{\theta}-\sin^2{\theta}}=\frac{1+\tan{\theta}}{1-\tan{\theta}}$$
Here's my attempt
$$\require{cancel}\text{Left - Right} = \frac{(1+2\sin{\theta}\cos... | Use
\begin{equation}
2 \sin \theta \cos \theta
=
\sin 2\theta
=
\frac{2 \tan \theta}{1 + \tan^2 \theta}
\end{equation}
\begin{equation}
\cos^2 \theta - \sin^2 \theta
=
\cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}
\end{equation}
Replacing the above quantities, we get
\begin{equation}
\frac{1 + \fr... | {
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"url": "https://math.stackexchange.com/questions/2887740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Value of a triangle inscribed in a rectangle I was doing a little trigonometry challenge given by my professor, which is not anything especially complicated. However, it is completely based on getting the area of the green triangle in relation to the side of the square and the triangle, equal to $x$.
Reference image: ... |
You can try this simpler alternative, which is in the spirit of finding the area of $\triangle GHB$ in terms of $x$.
Since we basically have a right triangle $ADE$, we can show that:
$$x^2+(2x)^2=DE^2\implies DE=\sqrt{x^2+5x^2}=x\sqrt5 \,\,\text{and:}\\
\sin \angle ADE=\frac{2x}{x\sqrt5}\implies \angle ADE=\arcsin\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2893767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Prove that $1 + \frac{2}{3!} + \frac{3}{5!} + \frac{4}{7!} + \dotsb = \frac{e}{2}$
Prove that $1 + \frac{2}{3!} + \frac{3}{5!} + \frac{4}{7!} + \dotsb = \frac{e}{2}$.
This is problem 4 from page 303 of S.L.Loney's 'Plane Trigonometry'.
It seems fairly obvious that the series expansion $e^x$ will be used. However, I ... | $$e=(1+1)+\left(\frac {1}{2!}+\frac {1}{3!}\right) +\left(\frac {1}{4!}+\frac {1}{5!}\right) +\left(\frac {1}{6!}+\frac {1}{7!}\right) +\cdots=2\left(1+\frac {2}{3!}+\frac {3}{5!}+\frac {4}{7!}+\cdots\right) $$
Q. E. D
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2894991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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find angle between two lines between $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6$ how to find to angle between these two lines $y-\sqrt{3}x-5=0$ and $\sqrt{3}y-x+6=0$
i tried so far like this
$y-\sqrt{3}x-5=0$
$y=\sqrt{3}x+5$
in the form of $y=mx+b$
got the value for $m_1=\sqrt{3}$
and for
$\sqrt {3}y-x+6=0$
$y=\dfrac {x-6}... | It's a very simple problem in essence. The gradients of the two lines are $\sqrt 3$ and $\frac 1{\sqrt 3}$. Recall that gradients equal the tangents of the angles made by lines with the $x$-axis. The special angles that give those tangents can quickly be recognised to be $60^{\circ}$ and $30^{\circ}$ (both in the first... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2896035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Is $ \sin 2x \ge 0 $ same as $\sin x\cos x\ge 0 $? Or differently phrased do $$ \sin 2x \ge 0$$ $$\sin x\cos x\ge 0 $$ have the same solution.
$\sin 2x$ is positive from $ 0\le x \le \pi/2$ , while $\sin x\cos x$ are positive in the first and third quadrant. Why is this, when $\sin 2x=2\sin x\cos x$, am i making ... | Note if $M > 0$ then $M*a*b > 0$ if $a > 0; b>0$ or $a < 0; b< 0$. And $M*a*b < 0$ if $a < 0; b > 0$ or $a > 0; b < 0$. And $M*a*b = 0$ if $a = 0$ or $b =0$. It does not matter whether $M =1$ or $M = 2$ or $M = \frac {37.95}{\pi}$.
So we would expect $\cos x \sin x \ge 0$ for the exact same values of $x$ as we would... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2896558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find unknown coefficient of a matrix with known rank. Let there be a matrix A = $\begin{bmatrix} 8 & 7 & 5 \\ 4 & 5 & 6 \\ 7 & 8 & λ \end{bmatrix}$.
Find the λ value, for which matrix A has a rank(A) = 2.
First I need to do the reduced row echelon form (I guess). Should I start by dividing the first row by 8 or by subs... | Hint: You need row $3$ to be a linear combination of rows $1$ and $2$. Write $a(8,7,5)+b(4,5,6)=(7,8,\lambda) $.
Solve the system $\begin{cases}{8a+4b=7 \\7a+5b=8}\end{cases}$ for $a$ and $b$.
Then $\lambda =5a+6b$.
If we let $M=\begin{pmatrix}8&4\\7&5\end{pmatrix}$, then $M^{-1}=\frac1{12}\begin{pmatrix}5&-4\\-7&8\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Farthest and Nearest Point in the ellipse from a focus using Derivatives Suppose I have an ellipse with major axis lying on the x- axis and center on origin $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$ and two foci at $(-c,0)$ and $(c,0)$.
How can I show that the farthest and nearest points from the right focus are $(-a,0)$... | As pointed out in the comments, $y'$ becomes infinite at the points you are looking for, causing your problems.
Instead, I would suggest you to to parameterize your ellipse as
$$\begin{cases}
x(t) = a\cos t \\
y(t) = b\sin t
\end{cases}, \quad 0 \leq t < 2\pi.$$
In this way you avoid any problems with differentiation.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2901840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
Given $a,b,c,d$ such that $a + b + c + d = 4$ show that $$(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$$
What I have tried so far is using CBS:
$(a^2 + 3)(b^2 + 3) \geq (a\sqrt{3} + b\sqrt{3})^2 = 3(a + b)^2$
$(c^2 + 3)(d^2 + 3) \geq 3(c + d)^2$
$... | We have
\begin{align}
(a^2+3)(b^2+3)
& =(a^2+1+2)(b^2+1+2) \\
& =(a^2+1)(b^2+1)+2(a^2+1+b^2+1)+4 \\
& = (a^2+1)(b^2+1)+2(a^2+b^2+2)+4 \\
& =(a^2+1)(b^2+1)+2(a^2+b^2)+8
\end{align}
By Cauchy Schwartz we obtain
$$(a^2+1)(b^2+1)\geq(a+b)^2\tag{$\star$}$$
$$2(a^2+b^2)=(1^2+1^2)(a^2+b^2)\geq(a+b)^2\tag{$\star \star$}$$
Puti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2902750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
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Where should I find the ranges for $\sum_{n=1}^k \sin n$ and other similar trigonometric series? It can be found that
$$\sum_{n=1}^k \sin n = \frac{\sin\left(\frac{k+1}{2}\right)\sin\left(\frac{k}{2} \right)}{\sin\left(\frac{1}{2}\right)},$$
$$ \sum_{n=1}^k \cos n = \frac{\cos\left(\frac{k+1}{2}\right)\sin\left(\frac{... | Note that $$\sin\left(\frac{k+1}{2}\right) \sin\left(\frac{k}{2}\right) = \frac{\cos(1/2) - \cos(k+1/2)}{2} \in \left[ \frac{\cos(1/2)-1}{2}, \frac{\cos(1/2)+1}{2}\right]$$
and
$$ \cos\left(\frac{k+1}{2}\right) \sin\left(\frac{k}{2}\right) = \frac{\sin(k+1/2)-\sin(1/2)}{2} \in \left[\frac{-1-\sin(1/2)}{2}, \frac{1-\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2904388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Partial Fraction Decomposition of quadratic factor I'm trying to break up the following equation into partial fractions:
$$\frac{1}{(x^2-1)^2}=\frac{1}{(x+1)^2(x-1)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+1}+\frac{D}{(x+1)^2}$$
I'm ending up with two equations and four unknowns though:
$$A+C=0 \\ -A + B + C+D = 0... | Alt. hint: $\;\dfrac{1}{x^2-1}=\dfrac{1}{2}\left(\dfrac{1}{x-1}-\dfrac{1}{x+1}\right)\,$, and therefore:
$$
\dfrac{1}{(x^2-1)^2}=\dfrac{1}{4}\left(\dfrac{1}{(x-1)^2}+\dfrac{1}{(x+1)^2}-\dfrac{2}{x^2-1}\right)=\ldots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2908080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the series converges and find its sum Show that
$$ \sum_{n=1}^\infty \left( \frac{1}{n(n+1)} \right) = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+ \;... $$
converges and find its sum.
My solution so far:
I am thinking about finding the partial sum first and show that the series converges since its finite partial ... | See the first two brackets. $-\frac{1}{2}$ in the first bracket and $\frac{1}{2}$ in the second bracket cancel each other out. This canceling happens all through out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Find the Matrix X and Y $\text { Find the matrix } X \text { and } Y , \text { if } X + Y = \left[ \begin{array} { l l } { 5 } & { 2 } \\ { 0 } & { 9 } \end{array} \right] \text { and } X - Y = \left[ \begin{array} { c c } { 3 } & { 6 } \\ { 0 } & { - 1 } \end{array} \right]$
Adding 1 and 2
$x + y +x-y = \left[ \begin{... | You have several errors. The most critical one is while calculating $y$:
$$y = \left[ \begin{array} { l l } { 5 } & { 2 } \\ { 0 } & { 9 } \end{array} \right] - \left[ \begin{array} { l l } { 4 } & { 4 } \\ { 0 } & { 4 } \end{array} \right] = \left[ \begin{array} { l l } { 1 } & { -2 } \\ { 0 } & { 5 } \end{array} \rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential equation of a vector field A vector field $X:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ is defined by the following formula:
$$ X
\begin{pmatrix}
x\\
y\\
z\\
\end{pmatrix}
= \begin{pmatrix}
-y(x^2+y^2)\\
x(x^2+y^2)\\
(1-x^2-y^2)
\end{pmatrix}$$
Determine formulas for the solution curves $y:\m... | So we essentially want to solve
$$ \begin{pmatrix}
\dot x(t)\\
\dot y(t)\\
\dot z(t)\\
\end{pmatrix}
= \begin{pmatrix}
-y(t)(x(t)^2+y(t)^2)\\
x(t)(x(t)^2+y(t)^2)\\
1-x(t)^2-y(t)^2
\end{pmatrix}$$
Let $x(t) = r(t)\cos\theta(t)$ and $y(t) = r(t) \sin\theta(t)$. I will omit the argument $t$ from now on an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find the length of the curve y = $\sqrt{-x(x+1)} - \arctan \sqrt{\frac{-x}{x+1}}$ I used the formula for this example:
$\displaystyle L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$
And I start by computing the derivative:
\begin{align*}
y' &= \left(\sqrt{-x^2-x}\right)' - \left(\arctan \sqrt{\frac{-x}{x+1... | For your actual task it might be beneficial to substitute $z = \sqrt{\tfrac{-x}{x+1}}$. Then your curve is $y = \tfrac z{1+z^2}-\arctan(z)$. Note that you will also have to adapt the boundaries.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Evaluate $ \lim_{x \to 0} \left( {\frac{1}{x^2}} - {\frac{1} {\sin^2 x} }\right) $ $$\lim_{x\to0}\left({\frac{1}{x^2}}-{\frac{1}{\sin^2x}}\right)$$
Using the L'Hospital Rule I obtained the value $-1/4$, but the answer is given to be $-1/3$. I can't find the mistake. Here's what I did; please point out the mistake.
\beg... | Your mistake probably comes from your third row, because the left limit does not exist and you may not apply L'Hospital there (and the other limit is $0$).
What you can do instead (notice the asymmetry):
$$\lim_{x\to0}\frac{\sin^2x-x^2}{x^2\sin^2x}=\lim_{x\to0}\frac{\sin^2x-x^2}{x^4}=\lim_{x\to0}\frac{\sin x+x}{x}\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 9,
"answer_id": 3
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Prove the inequality using Chebyshev's Inequality If $a,b,c \in(0,\infty)$, then prove that: $$9(a^3+b^3+c^3)\ge(a+b+c)^3$$
I was trying to prove this inequality using Chebyshev's Inequality and assuming $a\ge b \ge c$ but to no avail.
Can please someone help me in proving it using Chebyshev's Inequality?
| Since $(a,b,c)$ and $(a^2,b^2,c^2)$ they are the same ordered, by Chebyshov twice we ontain:
$$9(a^3+b^3+c^3)=3\cdot3(a^3+b^3+c^3)\geq3(a+b+c)(a^2+b^2+c^2)\geq(a+b+c)^3.$$
Also, by Holder
$$(1+1+1)^2(a^3+b^3+c^3)\geq\left(\sqrt[3]{1^2a^3}+\sqrt[3]{1^2b^3}+\sqrt[3]{1^2c^3}\right)^3=(a+b+c)^3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2910910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Let $n \in \mathbb{N}$ such that n is a non-perfect square. Show that $\sqrt{n}$ is irrational. Can someone please check my proof?
Let $n \in \mathbb{N}$ such that n is a non-perfect square. Show that $\sqrt{n}$ is irrational.
Let's prove it by contradiction, that is, suppose there are $n,a,b \in \mathbb{N}$ such that ... | Don't bother with even or odd cases. Just do:
$\frac {a^2}{b^2} = n$ so $a^2 = nb^2$. So for any prime, $p$ that divides $b$ then $p|a^2$ so $p|a$. That's impossible because $a$ and $b$ are coprime. So there are no primes that divide $b$ (!!!!!!). So $b = 1$. And $a^2 = n$ and $n$ is a perfect square.
That's it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove this inequality using AM-GM? Suppose $a,b,c$ are positive real numbers. Then prove that $$\Big(\frac{a+b}{2}\Big)\Big(\frac{b+c}{2}\Big)\Big(\frac{c+a}{2}\Big)\ge\Big(\frac{a+b+c}{3}\Big)\Big(abc\Big)^\frac{2}{3}\tag{*}$$
My approach: From AM-GM $$\Big(\frac{a+b}{2}\Big)\ge\Big(ab\Big)^\frac{1}{2}\tag{1}$$... | By AM-GM: $$\prod_{cyc}(a+b)-\frac{8}{9}(a+b+c)(ab+ac+bc)=\frac{1}{9}\sum_{cyc}(a^2b+a^2c-2abc)=$$
$$=\frac{1}{9}\sum_{cyc}(a^2c+b^2c-2abc)\geq\frac{1}{9}\sum_{cyc}(2\sqrt{a^2c\cdot b^2c}-2abc)=0.$$
Id est, by AM-GM again we obtain:
$$\prod_{cyc}\left(\frac{a+b}{2}\right)^3=\frac{1}{512}\left(\prod_{cyc}(a+b)\right)^3\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find nullspace of this matrix The question I have is this:
1) Find a basis for the solution space of the homogeneous linear system.
$$x - y + 2z = 0 \\
2x + y = 0 \\
x − 4y + 6z = 0$$
So I convert to a matrix:
$$\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & 0 \\ 1 & -4 & 6 \end{bmatrix}$$
$$ -> \begin{bmatrix} 1 & -1 & 2 \\ 0 ... | Looks to me like the right method to get the rational basis for the null space. Quick check also shows that
$$
\left[
\begin{matrix}
1 & -1 & 2 \\
2 & 1 & 0 \\
1 & -4 & 6 \\
\end{matrix}
\right]
\left[
\begin{matrix}
-2/3 \\
4/3 \\
1 \\
\end{matrix}
\right] =
\left[
\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Simple expression for $\sum_{n=0}^{\infty} {2n+E-1 \choose 2n} x^{2n}$? The title basically says it all. If $2n$ was just $n$, I think it's relatively simple:
$\sum_{n=0}^{\infty} {n+E-1 \choose n} x^{n} = \frac{1}{(1-x)^E}$.
But what if you want every other term, i.e.
$\sum_{n=0}^{\infty} {2n+E-1 \choose 2n} x^{2n}$... | We have
\begin{eqnarray*}
\sum_{n=0}^{\infty} \binom{n+E-1}{n} x^n =\frac{1}{(1-x)^E}.
\end{eqnarray*}
Break this into its odd & even powers of $x$
\begin{eqnarray*}
\sum_{n=0}^{\infty} \binom{2n+E-1}{2n} x^{2n} +\sum_{n=0}^{\infty} \binom{2n+E}{2n+1} x^{2n+1} =\frac{1}{(1-x)^E}=\frac{(1+x)^E}{(1-x^2)^E}.
\end{eqnarray... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2914746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solving $\sin(3x +28^\circ)= \cos(2x-13^\circ)$
Solve the following equation :
$$\sin(3x +28^\circ)= \cos(2x-13^\circ), x \in [0^\circ, 360^\circ]$$
Solution:
$$3x+28 +2x -13 =90$$
$$x=15$$
$$3x+28 -2x +13 = 90$$
$$x= 49$$
But the value $$x = 87$$ also satisfies the equation. I do not know how it comes.
| Alternatively:
$$\sin(3x +28^\circ)= \cos(2x-13^\circ) \Rightarrow \\
\sin(3x +28^\circ)= \sin(90^\circ -(2x-13^\circ)) \Rightarrow \\
\sin(3x +28^\circ)-\sin(103^\circ-2x)=0 \Rightarrow \\
2\cos \frac{3x+28^\circ+103^\circ-2x}{2}\sin \frac{3x+28^\circ-(103^\circ -2x)}{2}=0 \Rightarrow \\
\cos \frac{x+131^\circ}{2}\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to find a value for a variable that makes a matrix (with said variable) equal to its own inverse I'm given $$\begin{bmatrix}3&x\\-2&-3\\\end{bmatrix}$$
and am asked to find x such that it's inverse would equal itself. To attempt this I first tried to put the question into an augmented matrix and got this:
$$\begin{... | Both methods you attempted must give correct answer.
Method 1. Finding the inverse from the augmented matrix:
$$ \left[
\begin{array}{cc|cc}
1&0&3&x\\
0&1&-2&-3
\end{array}
\right] \stackrel{R_1/3}=
\left[\begin{array}{cc|cc}
\frac13&0&1&\frac x3\\
0&1&-2&-3
\end{array}
\right] \stackrel{2R_1+R_2\to R_2}=\\
\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2915113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Prove the inequality $\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$ Let $a;b;c\in R^+$ such that $ab+bc+ca>0$. Prove that $$\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$$
I have seen the similar question is $$\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\fra... | Because by AM-GM and C-S we obtain:
$$\sum_{cyc}\sqrt{\frac{a^2+1}{b+c}}\geq3\sqrt[6]{\prod_{cyc}\frac{a^2+1}{b+c}}=3\sqrt[12]{\prod_{cyc}\frac{(a^2+1)(1+b^2)}{(a+b)^2}}\geq3\sqrt[12]{\prod_{cyc}\frac{(a+b)^2}{(a+b)^2}}=3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2922166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve Recurrence relation: $T(n) = T(n/3) + T(n/5) + n$ Given the following: $T(n) = T(n/3) + T(n/5) + n$
The aim is to find the upper bound of this solution.
I have tried to solve this using Master's Theorem, but this does not apply for this problem.
My next best options would be Substitution or Iteration, but... | Let's check if
\begin{equation}
T(n) = \frac{15}{7}n
\end{equation}
is a solution for our recursion.
\begin{align}
T(n) &= \frac{15}{7}n \\
T(\frac{n}{3}) &=\frac{1}{3} \frac{15}{7}n\\
T(\frac{n}{5}) &=\frac{1}{5} \frac{15}{7}n
\end{align}
So
\begin{equation}
T(\frac{n}{3}) + T(\frac{n}{5}) + n
=
\frac{1}{3} ... | {
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"url": "https://math.stackexchange.com/questions/2923492",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Solution of a quadratic equation to satisfy a constraint I have the roots of a quadratic equation as
$$x = \frac{1 \pm \sqrt{1-4\theta^2}}{2\theta}$$ I know that $|\theta| < 1/2$. Among these two roots, I want to find the one which has a value $|x| < 1$.
My attempt is:
It can't be $\frac{1 + \sqrt{1-4\theta^2}}{2\theta... | Alt. hint: $\;x_1+x_2=1/\theta \gt 0\,$ and $\,x_1x_2=1 \gt 0\,$, so both $\,x_1,x_2 \gt 0\,$. Since $\,x_1x_2=1\,$ it follows that $\,0 \lt x_1 \lt 1 \lt x_2\,$, so the smaller root, corresponding to the "$-\sqrt{\Delta}$" sign, is within $\,(0,1)\,$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Partial decomposition of $\frac{z^2}{(2z^2+3)((s^2+1)z^2+1)}$ Recently I have come arcross the following fraction
$$\dfrac{z^2}{(2z^2+3)((s^2+1)z^2+1)}$$
Hence I have encountered this fraction within a task of integration I want to do a partial decomposition. First of all I rewrote it as following
$$\dfrac{z^2}{(2z^2... | Even though we could solve this using the method of residues, I'll follow your method
\begin{equation}
\frac{z^2}{(2z^2+3)((s^2+1)z^2+1)}
=
\frac{Az + B}{2z^2 + 3}
+
\frac{Cz + D}{(s^2+1)z^2 + 1}
\tag{1}
\end{equation}
which is
\begin{equation}
\frac{z^2}{(2z^2+3)((s^2+1)z^2+1)}
=
\frac{(Az + B)((s^2+1)z^2 + 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2929238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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For $a\geq2$, $b\geq2$ and $c\geq2$, prove that $\left(a^3+b\right)\left(b^3+c\right)\left(c^3+a\right)\geq125 abc$
For $a\geq2$, $b\geq2$ and $c\geq2$, prove that
$$(a^3+b)(b^3+c)(c^3+a)\geq 125 abc.$$
My try:
First I wrote the inequality as
$$\left(a^2+\frac{b}{a}\right) \left(b^2+\frac{c}{b}\right) \left(c^2+\frac... | You did 90% of the work. You can finish it by using a little bit of Calculus.
Let
$$f(x)=x^2+\frac{2}{x}.$$
Since the derivate of $f(x)$ is positive, the function $f(x)$ is increasing for $x\geq2$, and then
$$f(x)\geq f(2)=5.$$
Thereby,
$$a^2+\frac{b}{a}\geq 5, \\ b^2+\frac{c}{b}\geq 5,\\ c^2+\frac{a}{c}\geq 5,$$
which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2930755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Finding $\text{Res}(f,0)$ where $f(z)=\frac{1}{z^2\sin(z)}$ I am trying to determine the residue of $z=0$ where $f(z)=\frac{1}{z^2\sin(z)}$.
I have determined that $z=0$ is a pole of order $3$. Hence to compute the residue, I use $$\text{Res}(f,0)=\frac{1}{2}\lim_{z\to 0}\frac{\partial^2}{\partial z^2}\left(\frac{z}{\s... | $$\sin z=z-\dfrac{1}{3!}z^3+\dfrac{1}{5!}z^5-\dfrac{1}{7!}z^7+\cdots$$
then
\begin{align}
\dfrac{1}{z^2\sin z}
&= \dfrac{1}{z^3\left(1-\dfrac{1}{3!}z^2+\dfrac{1}{5!}z^4-\dfrac{1}{7!}z^6+\cdots\right)} \\
&= \dfrac{1}{z^3}\left(1+\dfrac{1}{6}z^2+(\dfrac{1}{36}-\dfrac{1}{120})z^4+\cdots\right) \\
&= \dfrac{1}{z^3}+\dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many words from $5$ different letters can be formed out of $10$ consonants and $5$ vowels if... How many words from $5$ different letters can be formed out of $10$ consonants and $5$ vowels if 'a' is always one of the vowels and the words have at least $2$ consonants?
My approach to this has been like this:
$1$ way... |
How many words with five different letters can be formed from ten consonants and five vowels if $a$ is always one of the vowels and the word contains at least two consonants?
Method 1: Consider cases, depending on the number of consonants. Since $a$ must be included, if you have at least two consonants, then the wor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2931662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find $a$ so that a line is tangent, secant or external from a sphere I am given the following problem:
Given the line $$r \{ R = (1,0,a) + \lambda [a \quad a \quad 0]$$ and the sphere $$S \{ 8x^2 + 8y^2 +8z^2 - 16x +24y -8z + 19 = 0$$ find, relating to values of $a$, when the line is external, tangent and secant to th... | let $u$ be the vector from $(1,0,a)$ to the center of the circle (1,-\frac 32, \frac 12)
$u = (0,-\frac 32, \frac 12-a)$
let $v$ be the direction vector of the line. $(a,a,0)$
$\frac {u\cdot v}{v\cdot v} v$ will give the projection of $u$ onto $v$
$u - \frac {u\cdot v}{v\cdot v} v$ will be orthogonal to $v$ and be the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2932095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How to solve following integral I am trying to solve following integral.
$$=\int_0^\infty\frac{x}{x^n+(\epsilon+\sigma)}dx$$
I started by assuming that $a = \epsilon + \sigma$ and $$\implies x^n = A\tan^2(\theta)$$
Then,
$$\implies x = a^{1/n}\tan^{2/n}(\theta)$$
$$\implies dx = \frac{2a^{1/n}}{n} (\tan(\theta))^{2/n-... | Observe we have
\begin{align}
\int^\infty_0 \frac{x}{x^n+a}\ dx = a^{(2-n)/n}\int^\infty_0 \frac{y}{1+y^n}\ dy.
\end{align}
Hence it suffices to evaluate
\begin{align}
\int^\infty_0 \frac{y}{1+y^n}\ dy.
\end{align}
The easiest way is to use contour integration. Observe
\begin{align}
\int^R_0 \frac{y}{1+y^n}\ dy + \int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2932746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Prove: $x^4 + y^4 + z^4 - 4(x^3 + y^3 + z^3) + 5(x^2 + y^2 + z^2) \leqslant 4 $ Prove that $$x^4 + y^4 + z^4 - 4(x^3 + y^3 + z^3) + 5(x^2 + y^2 + z^2) \leqslant 4$$ for all $x, y, z \geqslant 0$ satisfying $x + y + z = 2$. When does equality occur?
Here is my aproach:
For each $x\in [0,2]$ we have: $$x^4-4x^3+5x^2-2x... | The homogenization gives
$$\frac{1}{4}(x+y+z)^4-\frac{5}{4}(x^2+y^2+z^2)(x+y+z)^2+2(x+y+z)(x^3+y^3+z^3)-x^4-y^4-z^4\geq0$$ or
$$\sum_{cyc}(x^3y+x^3z-2x^2y^2+x^2yz)\geq0,$$ which is true by AM-GM or by Muirhead because $(3,1,0)\succ(2,2,0).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2933273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Finding $ \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$
Finding $\displaystyle \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$
Try: Let $$\displaystyle I = \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$$ (Function is even )
$$I = 2\int^{1}_{0}\bi... | Let us consider your integral:
$$I = \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$$
Separate the integral:
$$I= \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln (1+x)\ dx-\int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln (1-x)\ dx$$
Consider the right-hand integral. Let $x=-t$ such that $dx=-dt$, where $t\in (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2935264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can you simplify $\sqrt{9-6\sqrt{2}}$? How do you simplify: $$\sqrt{9-6\sqrt{2}}$$
A classmate of mine changed it to $$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$ but I'm not sure how that helps or why it helps.
This questions probably too easy to be on the Math Stack Exchange but I'm not sure where else to post it.
| Your class mate is being.... clever.
If $\sqrt {9-6\sqrt 2}=a-b $ then $9-6\sqrt 2=a^2-2ab+c^3$
Let $2ab=6\sqrt 2$ and $a^2+b^2=9$.
Can we do that?
If we let $b^2=k $ and $a^2=9-k$ then $ab=\sqrt {k (9-k)}=3\sqrt 2=\sqrt {18} $. Solving $k (9-k)=18$ for $k $ (if it isn't visiblely obvious that we can do it in our hea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2936269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Laurent expansion for $1/\cos(z)$ I have a quick question.
How to find the Laurent expansion for $1/\cos(z)$
In the link above the person asks how to find the Laurent expansion for $\frac{1}{cos(z)}$.
The accepted answer utilizes the fact that \begin{align}
\frac{1}{\sin t}&=\frac{1}{ t-\frac{t^3}{3!}+\frac{t^5}{5!}-\f... | Here is a simple justification:
$$1-\left(\frac{t^2}{3!}-\frac{t^4}{5!}+\frac{t^6}{7!}-\dotsm\right)=\frac{\sin t}t,$$
so that $\enspace\dfrac{t^2}{3!}-\dfrac{t^4}{5!}+\dfrac{t^6}{7!}-\dotsm=1-\dfrac{\sin t}t$, and it is well-known the cardinal sine function
$$\operatorname{sinc}(t)=\begin{cases}\dfrac{\sin t}t&\text{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find height of a wall if at the beginning it exceeds $10$ meters and then $8$ meters When the foot of a staircase is $5$ meters from the base of a wall, it protrudes $10$ meters above the wall; and if it is $9$ meters from the base, it stands $8$ meters.
Find the height of the wall.
Using the Pythagorean theorem for ... | Let $z$ be the length of the ladders, then
$$(z-10)^2=5^2+y^2$$
$$(z-8)^2=9^2+y^2$$
subtracting the two equations, we have
$$-2(2z-18) = (5-9)(5+9)$$
$$4(z-9) = (9-5)(5+9)=4(14)$$
$$z=9+14=23$$
$$(23-10)^2-5^2=y^2$$
$$y^2=13^2-5^2=12^2$$
Hence $y=12$.
Note my definition of $z$ is not the hypothenus but the length of l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determining the minimum value of the function $y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$ I am curious whether there is an algebraic verification for $y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$ having its minimum value of $\sqrt{2 + \sqrt{3}}$ at $\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$. I have been told the graph of it is that ... | Without derivatives,
Considering the functions
$$
f(x,y)=y-x-2\sqrt{x^2-\sqrt 2 x-1} = 0\\
y = \lambda
$$
Their intersection is at the solution for
$$
f(x,\lambda)=\lambda-x-2\sqrt{x^2-\sqrt 2 x-1} = 0
$$
or squaring
$$
(x-\lambda)^2-2(x^2-\sqrt 2 x-1)=0
$$
Solving for $x$ we have
$$
x = \frac{1}{3} \left(2 \sqrt{2}\pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2942263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
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Finding the complex square roots of a complex number without a calculator
The complex number $z$ is given by $z = -1 + (4 \sqrt{3})i$
The question asks you to find the two complex roots of this number in the form $z = a + bi$ where $a$ and $b$ are real and exact without using a calculator.
So far I have attempted to ... | Note that$$z=7\left(-\frac17+\frac{4\sqrt3}7i\right).\tag1$$Now, since $\left(-\frac17\right)^2+\left(\frac{4\sqrt3}7\right)^2=1$, the expression $(1)$ expresses $z$ as $7\bigl(\cos(\alpha)+\sin(\alpha)i\bigr)$, for some $\alpha$. So, a square root of $z$ is $\sqrt7\left(\cos\left(\frac\alpha2\right)+\sin\left(\frac\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 2
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Determine linear map so that equation is correct Let $f:\mathbb R^3→\mathbb R^2$, $(x,y,z)^T\mapsto(x^2y-5z,2x+4yz-3z^3)^T$,
and $(x_0, y_0, z_0)^T=(-1,0,1)^T$.
I need to determine the linear map $A∈L(\mathbb R^3,\mathbb R^2)\cong\mathbb R^{2×3}$ so that$$
f \begin{pmatrix} x \\ y \\ z \end{pmatrix}=f\begin{pmatrix} x_... | Guide:
Use Taylor series expansion.
Let $(x,y,z) = (x_1, x_2, x_3)$ and $f(x,y,z)=(w_1,w_2)$.
Compute $A_{ij} = \frac{\partial w_i}{\partial x_j}$ and evaluate at $(x_0, y_0, z_0)$.
Edit:
For checking, I just drop higher order terms.
\begin{align}
x^2y-5z &= [(x-1)+1]^2y-5z \\
&\approx [2(x-1)+1]y - 5z \\
&\approx y-5z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2944182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove divisibility of a number using the binomial expansion? I have the following problem:
Prove that $6^n-1$ is always divisible by 5 using the binomial expansion of $(5+1)^n$.
How can I do this? I don't know how to begin, as I don't see how the binomial expansion relates to the question. Any help would be ap... | $$6^n-1 = (5+1)^n-1$$
$$(5+1)^n = {n \choose 0}5^n+{n \choose 1}5^{n-1}+{n \choose 2}5^{n-2}+{n \choose 3}5^{n-3}+...{n \choose n-1}5^1+{n \choose n}1$$
$$(5+1)^n-1 = {n \choose 0}5^n+{n \choose 1}5^{n-1}+{n \choose 2}5^{n-2}+{n \choose 3}5^{n-3}+...{n \choose n-1}5^1+{n \choose n}1-1$$
${n \choose n}1 = 1$ so $1$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2944723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I complete the square of $y= -4x^2-2x-4$? $y = -4x^2 - 2x - 4$
I just can't figure this out, do I divide the second number and the third number by $4$ then by $2$ and then add the product to the second one and subtract it from the third one?
| In general, if you want to complete the square on $ax^2+bx+c$, here's the formula and derivation
Suppose that
$$ax^2+bx+c=A(x+B)^2+C$$
Then:
$$ax^2+bx+c=Ax^2+2ABx+AB^2+C$$
Therefore:
$$A=a$$
$$b=2aB$$
$$c=aB^2+C$$
Therefore:
$$B=\frac{b}{2a}$$
$$C=-a(\frac{b}{2a})^2+c=-\frac{b^2}{4a}+c$$
Plug it all in:
$$ax^2+bx+c=a(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2944950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does Wolfram|Alpha make a mistake here? We want to evaluate $$\lim_{x \to -8}\frac{\sqrt{1-x}-3}{2+\sqrt[3]{x}}.$$The solving process can be written as follows:\begin{align*}\lim_{x \to -8}\frac{\sqrt{1-x}-3}{2+\sqrt[3]{x}}&=\lim_{x \to -8}\left[\frac{(\sqrt{1-x}-3)(\sqrt{1-x}+3)}{(2+\sqrt[3]{x})(4-2\sqrt[3]{x}+\sq... | In Mathematica 11.3 I get
In[1]:= Limit[(Sqrt[1 - x] - 3)/(2 + CubeRoot[x]), x -> -8]
Out[1]= -2
Mathematica Documentation says CubeRoot[x] gives the real-valued cube root of $x$.
Even
In[4]:= -8^(1/3)
Out[4]= -2
Mathematica gives me the correct answers
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2946885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 3,
"answer_id": 1
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$\frac{a^{3}+1}{b+1}+\frac{b^{3}+1}{a+1}$ an integer $\Rightarrow \frac{a^{3}+1}{b+1}$ and $\frac{b^{3}+1}{a+1}$ are integers. I want to show that if the natural numbers $a,b \in \mathbb{N}$ are such that $\frac{a^3+1}{b+1}+\frac{b^3+1}{a+1} \in \mathbb{N}$, then, necessarily, $\frac{a^3+1}{b+1} \in \mathbb{N}$ and $\f... | Hint: You can prove this more general statement:
Let $w,x,y,z$ be positive integers so that
$$\frac{w}{x}+\frac{y}{z}\in\mathbb{N}$$
and $z|w,x|y$. Then
$$\frac{w}{x},\frac{y}{z}\in\mathbb{N}.$$
To do so, consider $d=\gcd(x,z)$, which must divide each of $w$ and $y$, so by dividing each variable by $d$ it suffices to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maximum and minimum absolute value of a complex number
Let, $z \in \mathbb C$ and $|z|=2$. What's the maximum and minimum value of $$\left| z - \frac{1}{z} \right|? $$
I only have a vague idea to attack this problem.
Here's my thinking :
Let $z=a+bi$
Exploiting the fact that, $a^2+b^2=4$
We get $z-\dfrac{1}{z}=a-\df... | Your idea is good, but you lose yourself in some computations (the maximum is correct, though).
Consider the square of the modulus:
$$
f(z)=\left|z-\frac{1}{z}\right|^2=\frac{|z^2-1|^2}{|z|^2}
$$
Since $|z|=2$ by assumption, we can as well consider
$$
g(z)=|z^2-1|^2=(z^2-1)(\bar{z}^2-1)=z^2\bar{z}^2-z^2-\bar{z}^2+1=5-z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 3
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How to integrate $1/(\sin x + a\sec x)^2$? How to integrate $1/(\sin x + a\sec x)^2$?
I tried by squaring the contents and trying to make use of some identities, but it became very messy and difficult to solve.
| $\displaystyle I(a) = \int \dfrac{\mathrm{d}x}{(\sin x + a\sec x)^2}$
Multiplying the numerator and denominator by $\cos^2x,$
$\displaystyle I(a) = \int \dfrac{\cos^2x\,\mathrm{d}x}{(\sin x\cos x + a)^2}$
$\displaystyle I(a) = \int \dfrac{\cos^2x\,\mathrm{d}x}{\sin^2x\cos^2x + 2a\sin x\cos x + a^2}$
$\displaystyle I(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $f$ be a function $f : \mathbb{N} \to \mathbb{N}$ such that $f(2x+3y)=f(x)f(y)$, determine $f$ here what I did .
$$f(0)=f(0)^2$$
so $f(0)=1$ or $f(0)=0$
IF $f(0)=1$
we have $f(2y)=f(y)$
$$f(1)=f(2)=\ldots=f(2^n)=a$$
the equation $f(x)-a=0$ has infinitly many solutions , so $f(x)=a$
since f(0)=1 , $f(x)=1$.
i don't ... | A priori, there is no reason to expect $f$ to be a polynomial. So knowing that $f(x) = 0$ for infinitely many $x$ doesn't mean that $f$ is identically zero.
I consider the case when $f(0) = 1$. Then $f(2x + 0y) = f(x)$, so $f(x) = f(2x)$ for all $x$. Similarly, $f(x) = f(3x)$ for all $x$.
Let's evaluate $f(1)$.
Note th... | {
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"url": "https://math.stackexchange.com/questions/2951724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Roots of polynomial equation $x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0$
If $x_1,x_2,...,x_6$ be the roots of $x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0$ then I have to show that $|x_j|=2\space\space\space\forall j\in\{1,2,3,4,5,6\}$
I get that the roots of the equation must be complex, of the form $a+ib$ where $|a+ib|=\sqrt{a^2+b^2... | Hint:\begin{multline}x^6+2x^5+4x^4+8x^3+16x^2+32x+64=\\=64\left(\left(\frac x2\right)^6+\left(\frac x2\right)^5+\left(\frac x2\right)^4+\left(\frac x2\right)^3+\left(\frac x2\right)^2+\frac x2+1\right).\end{multline}Also, use the fact that $x^7-1=(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Sum of $\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - ...$ I have to find the sum of :
$$\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - \frac{x^5}{5*4} +\cdots$$
So far I have :
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \, x^{n+1}}{(n+1)(n)}$$
which is very close to $\ln(1+x)$... but I just can't figure o... | Hint:
On its domain of convergence, the derivative of the sum of this power series is
$$x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\dotsm=\ln(1+x).$$
| {
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Problems with an exact differential eqution Consider the following differential equation
$$
\left(\frac{1}{x}-\frac{y^2}{(x-y)^2}\right)dx=\left(\frac{1}{y}-\frac{x^2}{(x-y)^2}\right)dy
$$
I want to find its general solution. I get that this equation is exact but, I trying to solve it for this method and it seems not... | The differential is indeed exact
$$\left(\frac{1}{x}-\frac{y^2}{(x-y)^2}\right)dx-\left(\frac{1}{y}-\frac{x^2}{(x-y)^2}\right)dy=0$$
Note that
$$\frac {dx}x=d\ln (x)$$
$$\frac {dy}y=d\ln (y)$$
And also that
$$
\begin{align}
E=&-\frac{y^2}{(x-y)^2}dx+\frac{x^2}{(x-y)^2}dy \\
E=&\frac{-y^2dx+x^2dy}{(x-y)^2}\\
E=&\frac{-y... | {
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A theorem for calculating the perpendicular diagonals' slopes of a quadrilateral
If the slopes of side $AB$, $BC$, $CD$, $DA$ of a quadrilateral $ABCD$ are respectively $m_1, m_2, m_3, m_4$, and the diagonals (or the extended lines of them) are perpendicular, then the equation in $p$:
$$(m_1m_3-m_2m_4)p^2-((m_2+m_4)... | $\def\peq{\mathrel{\phantom{=}}{}}$Since the equation remains the same under translation, without loss of generality assume that line $AC$ and $BD$ intersect at the origin $O$. Denote by $p$ the slope of $AC$ (Assume that it exists).
Case 1: $p = 0$. In this case, $y_A = y_C = 0$, and $AC ⊥ BD$ implies $x_B = x_D = 0$.... | {
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Parametric equation and distance formula Help with this assignment! For the past three weeks now, I've been battling with mathematics assignment. I've successfully solved some but these ones posed a great challenge to me
*
*Eliminate the parameter and identify the graph of the resulting equation
$x = 3v+7, y = v-1... | $x = \sin t, y = \cos t, 0\leq t \leq \pi$
I would really hope that you just recognize this as a circle, as this is close to the unit circle definition of cosine and sine.
You could say:
$x^2 = \sin^2 t\\ y^2= \cos^2 t\\
x^2 + y^2 = \sin^2 t + \cos^2 t = 1$
Considering the limits on $t$ once that is applied, you will... | {
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Solving $2 \sin x \left( \sqrt{3} \cos x- \sin x \right)= \sqrt{2} - 1$ I wanted to know if this equation could be solved any further please.
$$2 \sin x \left( \sqrt{3} \cos x- \sin x \right)= \sqrt{2} - 1$$
I have gone this far:
$$4 \sin x \sin(60^\circ-x)= \sqrt{2}- 1$$
Thank you
| Hint: $2\sin x(\sqrt{3}\cos x - \sin x) = \sqrt{2} - 1\implies \sqrt{3}\sin(2x) - (1- \cos(2x)) = \sqrt{2} - 1\implies \sqrt{3}\sin(2x)+\cos(2x)=\sqrt{2}\implies \sin(2x+\frac{\pi}{6})= \sin(\frac{\pi}{4})$. At this point, can you continue to the finish line ….?
| {
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Why this $12b^2<49ac< \frac{49}{4}b^2$ inequality given ? What is the use of it? Question :
Let $\alpha$ and $\beta$ be roots of $ax^2+bx+c=0$. Show that
$(4\alpha-3\beta)(4\beta-3\alpha)=\frac{49ac-12b^2}{a^2}$. If
$12b^2<49ac< \frac{49}{4}b^2$ , then show that $\beta $ lies between
$\frac{3}{4}\alpha$ and $\f... | $$(4\alpha-3\beta)(4\beta-3\alpha)=25\alpha\beta-12(\alpha^2+\beta^2)=49\alpha\beta-2(\alpha+\beta)^2=$$
$$=\frac{49c}{a}-\frac{12b^2}{a^2}=\frac{49ac-12b^2}{a^2}.$$
$49ac<\frac{49b^2}{4}$ says that our equation has two real roots.
| {
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Domain and range of $g(x) = \frac{3x^2+4}{x-2}$
$$g(x) = \frac{3x^2+4}{x-2}$$
for the domain, I find, $\{x|x\ne 2\}$'
and for the range
$y = \dfrac{3x^2+4}{x-2}$ and when $x=2-\epsilon$, $y \to \dfrac{3(2-\epsilon)^2 + 4}{(2-\epsilon)-2} = \dfrac{K}{-\epsilon} \to -\infty$
When $x = 2+\epsilon$, $y \to \dfrac{3(2+\ep... | The domain is $$\mathbb R\setminus\{2\}.$$
Now, let $x>2$.
Thus, by AM-GM
$$\frac{3x^2+4}{x-2}=\frac{3(x-2)^2+12(x-2)+16}{x-2}=3(x-2)+\frac{16}{x-2}+12\geq$$
$$\geq2\sqrt{3(x-2)\cdot\frac{16}{x-2}}+12=12+8\sqrt{3}.$$
The equality occurs for $3(x-2)=\frac{16}{x-2}$ or $x=2+\frac{4}{\sqrt3}.$
Also, let $x<2$.
Thus, by AM... | {
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"timestamp": "2023-03-29T00:00:00",
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Multiple divisions I was confused a bit with a little arithmetic here. For instance $1÷1÷2$ and $2÷3÷7$. BODMAS isn't effective in this case. My question is this:
$2÷3÷7$
Am I to divide $2/3$ by $7$ or divide $2$ by $3/7$??
| You need to carry on first division first.
$$\frac{\frac{2}{3}}{7}=\frac{\frac{2}{3}}{\frac{7}{1}}=\frac{2}{3}\cdot\frac{1}{7}=\frac{2}{21}$$
while
$$\frac{2}{\frac{3}{7}}=\frac{\frac{2}{1}}{\frac{3}{7}}=\frac{2}{3}\cdot\frac{7}{1}=\frac{14}{3}$$
BODMAS is an acronym and it stands for Bracket, Of, Division, Multiplicat... | {
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$7^x-3^y=4$ in the integers I have to find all solutions to $7^x-3^y=4$ in the integers, I've already proven that $x$ and $y$ have the same parity and that they cannot be even. But I'm stuck in the case when $x$ and $y$ are odd. Could someone show me how to solve for all solutions? I know there is at least one, $(x,y)=... | If $y\leq 1$, the only solution is $(1,1)$, which you found. If $y\geq 2$, then consider the equation $\bmod 9$. We see
$$7^x\equiv 4\bmod 9$$
We have
$$7^0\equiv 1,\ 7^1\equiv 7,\ 7^2\equiv 4,\ 7^3\equiv 1,$$
so $7^x\equiv 4\bmod 9$ iff $x\equiv 2\bmod 3$. As a result, we have $x\equiv 5\bmod 6$ (as you have already s... | {
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