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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.

Without solving the quadratic equation find $x_1^2 - x_2^2$

Here's the full text of the problem: X1 and X2 are the solutions of the quadratic equation, without solving the equation find x1^2-x2^2.Read more

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Is $\ x^2 = 9\ $ equivalent to $\ x =\sqrt 9\ $?

$Is $\ x^2 = 9\ $ equivalent to $\ x =\sqrt 9\ $?$

The problem here is mainly notational. There is only one solution to x= √ 9 , which comes from the way that the square root function is defined.Read more

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Factoring $x^{7} - 1$ into irreducibles over $\mathbb{F}_{2}[x]

$Factoring $x^{7} - 1$ into irreducibles over $\mathbb{F}_{2}[x]$

2 Answers 2 ... Factoring x7−1 is equivalent to factoring x8−x. Now, x8−x=0 is the equation that defines the finite field with 8 elements.Read more

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General solution of $\sin(x) = -1/2

$General solution of $\sin(x) = -1/2$

There is no difference, since3π+(−1)37π6=11π6and3π+(−1)311π6=7π6. So, you get the same solutions in both cases.Read more

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algebra precalculus - How to solve $x^3 = 1

Take the cube root of both sides to identify a solution, and then divide (maybe with long division) by (x−one solution) to simplify the expression.Read more

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Solving trigonometric equation $\sin(2x) = \frac{1}{2}$

$Solving trigonometric equation $\sin(2x) = \frac{1}{2}$$

Method 2 always took one solution 2x=30 and divided by 2 to get x=15. At this point x=W2 is not the angle that was entered into sin expression.Read more

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Factor $x^4 + 64 - algebra precalculus

In this case we have b=2, so x4+64=(x2+4x+8)(x2−4x+8).Read more

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What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}, x \in \ ...

$What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}, x \in \ ...$

You basically used the formula a+b=a2−b2a−b. in an indirect way. The problem gave you a−b and, since a,b are square roots of simple ...Read more

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Convert 1-5 Grading Scale to 1-100 Grading System

It looks like very decrease of 3 points on the 1-100 scale gives an increase of .25 on the 1-5 scale. For example, going from 94-96 to 91-93 (a ...Read more

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linear algebra - Solve the system (x1, x2, x3)

By the looks of it, you've let x3 be a free variable s. Hence, in addition to x1=−10+13s and x2=11−17s, you also have x3=0+1s.Read more

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Integrate $\int \frac {\ln x}{x^2}$ by U-Substitution

$Integrate $\int \frac {\ln x}{x^2}$ by U-Substitution$

I'm not sure why starting with a substitution, since you have to use integration by parts anyway. However, you should remove x altogether, ...Read more

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Is there an identity for cos(ab)?

For general a and b, we cannot write cos(ab) in terms of the trig functions cosa,sina,cosb,sinb. This is because the trig functions are periodic ...Read more

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cot^{-1}x = \tan^{-1}\dfrac{1}{x}$ Why are my methods of ...

$cot^{-1}x = \tan^{-1}\dfrac{1}{x}$ Why are my methods of ...$

The easiest way to see that cot−1x is not always equal to tan−1(1/x) is to note that the inverse cotangent is always positive, while the inverse ...Read more

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Find $\int \sin^4(x)\cos^5(x)dx

$Find $\int \sin^4(x)\cos^5(x)dx$

22 Sept 2021 — Yes. Note thatsin4(x)cos5(x)=sin4(x)(1−sin2(x))2cos(x). So, do sin(x)=y and cos(x)dx=dy.Read more

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Why is ln(1−x)≈−x when x is small? [duplicate]

Because ln(1)=0 and ddtln(1−x)|t=0=−1. Therefore, when x is small, ln(1−x) is approximately 0+(−1)×x=−x.Read more

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Taylor series of $\ln(1+x)

$Taylor series of $\ln(1+x)$

Notice that for all x∈[0,1[, we have ln(1+x)=x∫011+tdt and for all x∈]−1,0], we have ln(1+x)=−0∫x11+tdt. (Note that the function t↦±11+t is ...Read more

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Solve $\sin(3x)=\cos(2x)

$Solve $\sin(3x)=\cos(2x)$

Question: Solve sin(3x)=cos(2x) for 0≤x≤2π. My knowledge on the subject; I know the general identities, compound angle formulas and double angle ...Read more

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Is $x^{1/2}$ equal to a positive value only?

$Is $x^{1/2}$ equal to a positive value only?$

My friend argues that √x is unequal to x1/2 because √x can equal both negative and positive values, whilst x1/2 can equal postive values only.Read more

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Why is the solution to $x-\sqrt 4=0$ not $x=\pm 2$?

$Why is the solution to $x-\sqrt 4=0$ not $x=\pm 2$?$

Well, x2−4=0⟹x2=4⟹x=√4. If the last implication is wrong, and should be x=±√4, then it might be a good idea to point it out at this point.Read more

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Solve: $\tan2x=1 - algebra precalculus

$Solve: $\tan2x=1 - algebra precalculus$

Your solution is not very rigorous after 0=tan2x+2tanx−1. After that line, the equation should be tanx=−2±√22−4(1)(−1)2(1).Read more

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Arc Length of $x^{\frac{2}{3}}+y^{\frac23}=1

$Arc Length of $x^{\frac{2}{3}}+y^{\frac23}=1$

11 Apr 2012 — If we assume that x2/3:=(x2)1/3 then the solution set of x2/3+y2/3=1 is symmetric about the x axis (since if (x,y) is an element of the solution ...Read more

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algebra precalculus - Factoring $x^4 - 16

Why is the factorization of x4−16=(x2+4)(x+2)(x−2) rather than (x2−4)(x2+4)?Read more

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Why is $y=1/x$ a continuous function but not $y=(1/x)^2$?

Therefore, y=1x is a continuous function because x=0 is not part of its domain. However, there is an infinite discontinuity at x=0, but because ...Read more

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Coincidence of 1/5/3 Being Equivalent to 3/5?

I believe you mean 1/(5/3); as you've written it, it's (1/5)/3, which is equal to 1/15. And yes, it is a very general rule that 1/(p/q) = q/p; ...Read more

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How would you solve for all solutions of $\sin(2x)=\cos(3x) ...

$How would you solve for all solutions of $\sin(2x)=\cos(3x) ...$

26 Mar 2015 — sin(2x)=cos(3x)⇒cos(π2−2x)=cos(3x)⇒π2−2x=±3x+2kπ,k∈Z. Can you take it from here?Read more

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General Solution of sin(2x)-cos(2x)=0 [closed]

General Solution of sin(2x)-cos(2x)=0 [closed] This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting ...Read more

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trigonometry - $ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

$trigonometry - $ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $$

I could start a semi-proof by using limits: as A→0,cosAcosBcosC becomes big (as we want), but A+B+C becomes too small. Also, as A→π2 ...Read more

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Prove trigonometry identity for $\cos A+\cos B+\cos C$

$Prove trigonometry identity for $\cos A+\cos B+\cos C$$

Let's first apply your observation to sin(C/2), along with the angle difference and sum formulas for sine, and the angle sum formula for cosine.Read more

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Is cos(x)×cos(2x) the same as cos(3x)

When you multiply cos(x)×cos(2x) the same as cos(3x) or do you have to treat each differently?Read more

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polynomials - Find all roots of $x^{6} + 1

$polynomials - Find all roots of $x^{6} + 1$

29 Mar 2014 — Hint: if x6=−1, then |x|6=1 and you can write x=cosθ+isinθ. details: Then the equation is, thanks to De Moivre theorem and cos2+sin2=1, ...Read more

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How to factorise $x^6 + 1

15 Apr 2013 — The fully factorized form of x6+1 (over the complex numbers) is therefore (x−i)(x+i)(x−12√3−12i)(x+12√3−12i)(x−12√3+12i)(x+12√3+12i).Read more

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Proving trigonometric Identity: $\frac{1+\sin x}{\cos x} = ...

$Proving trigonometric Identity: $\frac{1+\sin x}{\cos x} = ...$

14 Oct 2012 — We'll multiply 1+sinx+cosx1−sinx+cosx with the conjugate of the denominator. We'll do this from right to left, i.e. we pick the RHS and walk ...Read more

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Integration of $\frac{1}{x\ln(x)}$ by parts

$Integration of $\frac{1}{x\ln(x)}$ by parts$

I know that ∫dxxln(x)=ln(lnx)+C. However, I often use integration by parts and I wonder what is wrong here, and what should I do to avoid it ( ...Read more

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Prove that the order of $U(n)$ is even when $n>2

25 Jul 2014 — If U(n) has an element of even order, then the order of U(n) is even by Lagrange's Theorem. On the other hand, for n>2, the order of n−1 in U(n) is 2.Read more

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sin(3x) = \sin(x) - trigonometry

$sin(3x) = \sin(x) - trigonometry$

I know I'm supposed to do sin(3x)−sinx=0 but beyond that I'm stuck.. I tried expanding sin(3x) but that didn' ...Read more

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inequality - Much less than, what does that mean?

"Much less than" is a qualitative assessment of comparative inequality. a≪b means that a is not only less than b, but muchly so by some convention.Read more

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Simplifying $\sin^2 (\alpha) \cos^2 (\alpha)

$Simplifying $\sin^2 (\alpha) \cos^2 (\alpha)$

You can for example use Fourier transforms together with the convolution theorem to solve it and get Dr. Sonnhard Graubner's answer.Read more

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algebra precalculus - y = x^25 -- How to solve for x.

Yes, that is correct. Note that then we would have x25=(25√y)25=y as expected.Read more

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The roots of $x^n - 1=0

7 Nov 2017 — Obviously, x=1 is one of the roots. There must be other roots who are complex number. I guess they are related to exponent. I don't know how to get them and ...Read more

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abstract algebra - How to find roots of $X^5 - 1

The first factor gives you the obvious solution x=1. To find other roots, we need to solve the equation x4+x3+x2+x+1=0.Read more

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Why the limit of sin(x)x as x approaches 0 is 1? [duplicate]

Note that you need a rigorous definition of sin(x) before you can hope to have a rigorous proof that limx→0sin(x)/x=1.Read more

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Using the definition of a limit to prove 1/n converges to zero.

The sequence an you gave is an=1n, and the limit α is α=0. Therefore, we wish to prove that for any ε>0 there is a positive integer ...Read more

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Calculating the limit $\lim((n!)^{1/n})

$Calculating the limit $\lim((n!)^{1/n})$

You could use the fact that for a sequence (an) of positive terms, if limn→∞an+1an exists, then so does limn→∞n√an and the two limits are equal.Read more

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Why does $(-2^2)^3$ equal $-64$ and not $64

So for it to actually be 64, would the problem need parentheses around the -2? So would it have to be ((−2)2)3 for it to equal 64?Read more

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Prove $\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}

$Prove $\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$

trigonometry - Prove $\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$ - Mathematics Stack Exchange.Read more

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Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\ ...

$Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\ ...$

Then sinA, sinB, and sinC are actually the lengths of the sides opposite that three angles---that's essentially the law of sines. The area of ...Read more

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Sum of series $\sin x + \sin 2x + \sin 3x + \cdots

$Sum of series $\sin x + \sin 2x + \sin 3x + \cdots$

The series does not converge for all x. There are some x, for instance x=0 or x=π, for which the series converges to 0.Read more

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Is $\cot(x)=\frac{1}{\tan(x)}

$Is $\cot(x)=\frac{1}{\tan(x)}$

3 Apr 2016 — This equation merely indicates that cotx and 1tanx are equal in value (when they are both defined, of course). It does not say that cotx and 1 ...Read more

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How do I prove or disprove that the equation - cos - ( - sin - x

How do I prove or disprove that the equation cos(sinx)=1/x has only one real root? It's easy to do graphically, but I'm more interested in the ...Read more

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