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Monday, 11/05/2026

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Q&A for people studying math at any level and professionals in related fields.

Is there a limit of $\cos(n!)

$Is there a limit of $\cos(n!)$

By the periodicity of cos, this is just cos(2π∑i≥n+1di(n+1)(n ... What is limn→∞(12+12cos(πn!Read more

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Limit of the sequence $2^n|\cos n

$Limit of the sequence $2^n|\cos n$

Limit of the sequence 2n|cosn| · The limit can't be ∞ because there are infinitely many negative numbers in this sequence. · Hint: infinitely many ...Read more

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What is $\sin(\cos(x))

$What is $\sin(\cos(x))$

As already said in comments and answers, you cannot do much with embedded trigonometirc functions except developing as Taylor series.Read more

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Showing that $\displaystyle\lim_{n\rightarrow \infty} (4^{n}

$Showing that $\displaystyle\lim_{n\rightarrow \infty} (4^{n}$

I can think of a number of ways to prove that limn→∞(4n−n4)=∞ but none of them brief. Does anyone have a short proof of this? Thanks! calculus ...Read more

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What is the limit of the sequence n!/4^n? [duplicate]

Closed 8 years ago. I am trying to find the limit of the sequence by using root test and I don't understand why the limit is not zero? (the ...Read more

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problem solving - What are solutions to $2^x=x$?

There are no solutions, until you specify what x is (natural number, real number, complex number, matrix, etc)Read more

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What is $\cos^2(x)$ in relation to $\sin(x)

$What is $\cos^2(x)$ in relation to $\sin(x)$

Are you familiar with what is perhaps the most basic relationship between sine and cosine? For any x, cos2(x)+sin2(x)=1.Read more

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polynomials - How can I give the solutions of $\ x^5-7

$polynomials - How can I give the solutions of $\ x^5-7$

2 May 2017 — Do you mean x5−7=0? That would be a variation of the roots of unity.Read more

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Solve for x where sin(x)cos(x)= 2

Solution does exist: answer is about x=2.665+2πn. sin(x)=1 won't satisfy equation, so sin(x)<1. Let sin(x)=2t, t<0. Then 2tcos(x)=2, cos(x)=1t<0 ...Read more

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Why does the double-angle tangent formula give a different ...

We have, by the double angle formula, tan180∘=2tan90∘1−tan290∘, which means 0=2(∞)1−(∞)2. Isn't that an indeterminate form, because the ...Read more

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Triangle inscribed inside a circle: prove that abc=4×area×R

Let the sides (of triangle ABC) have lengths a,b,c as usual. The question suggests that we use some relationship of the form abc=constant×R. ...Read more

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Why does AB+BC=AC when adding vectors?

→AB is a vector while AB is a distance. Vectors verify the additive relation: →AB+→BC=→AC.Read more

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Simplification of: AB + A'C + BC in boolean algebra

The two expressions are not equal. The first expression is true when A and B is true and C false but the second is false in this case.Read more

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arithmetic - get 100 with the help of five 2

First, you have to group the twos into numbers. Five twos (2,2,2,2,2) can't give more than 32, so it's not enough.Read more

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Solution set for trigonometric equation sin(3x)+cos(3x)=0 ...

Your solutions for questions 17, 19, and 20 are correct. As for question 18: sin(3x)+cos(3x)=0sin(3x)=−cos(3x)tan(3x)=−13x=−π4+kπ ...Read more

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How to solve $0=\cos{2x}+\cos x - trigonometry

$How to solve $0=\cos{2x}+\cos x - trigonometry$

Yes, you are on the right track. I would expand cos2x=2cos2x−1, then put c=cosx and solve the resulting quadratic equation in c.Read more

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Why doesn't 1/x=0 have any solution? [closed]

The equation 1/x=0 is meaningful even if we are assuming that x is a (nonzero) real number, in which case it has no solution. The equation may ...Read more

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Why is $\sin^2x + \cos^2x = 1$ important?

$Why is $\sin^2x + \cos^2x = 1$ important?$

sin2x+cos2x=1 is basically just the Pythagorean identity (a2+b2=c2) expressed in Trigonometric terms instead of Algebraic terms. And that's ...Read more

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In how many different ways can you prove that $\sin^2x + \ ...

$In how many different ways can you prove that $\sin^2x + \ ...$

In how many different ways can you prove that sin2x+cos2x=1 [duplicate] ; 7. It all boils down to the definition of sin and cos. Which one are ...Read more

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calculus - Is (x+1/4) equivalent to x/4?

Or does x turn into x/1 + 1/4, and then I multiply x * 4/1 * 4 to get 4x/4 + 1/4 which would equal to 4x+1/4?Read more

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Find the last two digits of $2015^{2016}-2017

$Find the last two digits of $2015^{2016}-2017$

let n=20152016−2017. If you reduce everything modulo 100 you'll get n≡152016−17(mod 100). To reduce 152016 , we calculate 152=125≡25(mod 100).Read more

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Computing the last 4 digits of $2013^{2014^{2015}}

$Computing the last 4 digits of $2013^{2014^{2015}}$

This is an old exam question from the introduction to abstract algebra course I took 7 years ago. I tried using the Chinese Remainder Theorem. I ...Read more

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Finding the roots of $x^5-2$. My question is about ...

Well, you can think of 2 as 2e2kπi, with k∈Z. Then if x5−2=0 we have x5=2e2kπi. and so (x5)1/5=(2e2kπi)1/5. This way x=5√2e2kπi5.Read more

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trigonometry - Solve $\cos(x^o-60^o) = 0.6$

$trigonometry - Solve $\cos(x^o-60^o) = 0.6$$

19 Feb 2019 — Just taking the inverse cosine of the result is the "naive" way of solving the equation because it reveals only one solution out of an ...Read more

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Please explain where I made a mistake in this example ...

My solution: cos2(x)−cos(2x)=14. cos2(x)−(2cos2(x)−1)=14. cos2x−2cos2(x)+1=14. −cos2x=−34. cos2x=34. cos(x)=√32cos(x)=−√32.Read more

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Let $G$ be a group. Show that $\forall a, b, c \in G$, the ...

$Let $G$ be a group. Show that $\forall a, b, c \in G$, the ...$

Let G be a group. Show that ∀a,b,c∈G, the elements abc,bca,cab have the same order.Read more

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How to factor$ x^2+2xy+y^2

This is simply factoring (x+y) from each term. If it isn't clear how to do try, first try reversing the step by distributing the first sum in (4) ...Read more

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