Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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32 views

Verifying the cosine rule

Verify the following system of linear equations in cos A, cos B , and cosC. Triangle cannot be shown. Then use Cramer’s Rule to solve for cosC , and use the result to verify the Law of Cosines: ...
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1answer
22 views

How many gaussian integers have modulus 5?

My work so far: $$0+5i\quad 5+0i\\ 3+4i\quad 4+3i$$ so there are 4 such integers. But when I type it in Brilliant I get wrong!! Maybe for them 0 is not an integer so I type 2 and I get wrong! What the ...
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3answers
38 views

What is the best way to find the roots?

In Calculus, mainly when we compute areas we face equations like $$x^\frac{1}{2}=x^2$$ I know that I can take the square root of the both sides. Does anyone know another way to find the roots of the ...
0
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1answer
28 views

Hint finding exact value of half-angle when $\tan (\theta) = {3}$

Unlike others I've tried, I'm having a hard time with this half-angle exercise: If $tan(\theta)={3}$ and $\theta$ is in QIII, find $\tan\left(\frac{\theta}{2}\right)$ Here's what I know (or think I ...
2
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6answers
195 views

Algebra: What does “is defined for” mean?

In algebra what does: "Is defined for" mean? I have a question posted: $\sqrt{a+b}$ is defined for $-b \leq a$. The question posed is: Is this true... My question: WHAT DOES "Is Defined For" ...
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1answer
8 views

System of inequalities. Points of intersection?

$x^2+y^2<=81$ $y<x$ Is this correct? My answer: (-9sqrt(2)/2,-9sqrt(2)/2), (9sqrt(2)/2,9sqrt(2)/2)
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3answers
58 views

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$

If $ f(x)=x^2-3x+1$ then $ f(x-2) = ?$ I'm not sure how to properly deal with this function and solve for $f(x-2)$.
1
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1answer
20 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
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4answers
36 views

Prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$

How to prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$ without using calculus just trigonometry?
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0answers
14 views

What would be a formula to this “ trains ” meet problem?

Guy riding a bicycle with speed 9km/h leaves City A. One hour and 15 min later, a bike rider is leaving City B traveling towards City A with speed of 21km/h. At what distance/km would these two ...
4
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1answer
71 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
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2answers
20 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
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5answers
79 views

How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
2
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1answer
48 views

Bombelli's wild thought of cubic equations

In many books, like Visual Complex Analysis. talk about the real original of complex number. the author begin with this equation: $$x^3=15x+4$$ Then the author use the formula ...
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1answer
16 views

Intersection of graphs, and no solution for trig functions.

All I know is the c=asin(x-b) I don't know how to check the values of b for 'no solutions,' in the case of trig functions. Can someone people provide an algebraic method to solve this.
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0answers
52 views

Proof for A majorizes B

$\alpha = [\alpha_i] \in\mathbb R^n$ and $\beta = [\beta_i]$ where $\beta_1 = \beta_2 = ......=\beta_n = \frac{1}{n}\sum\alpha_i$ How can i show that $\alpha$ majorizes $\beta$ I tried to get a ...
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1answer
14 views

Choosing values for octave/decade scale

I am developing a frequency response simulator for linear circuits which should be able to plot graphs of Voltage x Frequency with the latest varying linearly, in octaves or in decades. The only ...
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2answers
41 views

Can this system be solved algebraically?

$$6x+zy=57\\ (4+x)(6+y)=81\\ z(6+y)=81$$ Is there a way to solve it algebraically? I already know the answer, I just need to know if it can be solved.
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1answer
31 views

question about rational expressions

i can't understand how to do this 5 question please help me ! Thank you
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1answer
42 views

Solve $\log_2 (1+\frac{1}{x-1})<1$

I don't get how my teacher got two different equations out of the one. One is $> 0$ and the other one is $<2$. Be detailed please.
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3answers
40 views

Sequence problem.

Help for this problem would be much appreciated, as I am expected to solve it without being properly taught how. Suppose a single cell of bacteria divides into three every 12 hours. Suppose that the ...
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3answers
21 views

Mean of 4 numbers

If the mean of 4 different integers is 75 and the largest of them is 90, what could be the least possible value of the smallest integer?
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1answer
25 views

Extraneous solutions where they come from?

I was doing some homework on logarithmic equations, and when I check my solutions on wolfram alpha I get that some aren't. So I'm interested in where do those extraneous equations come from?
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4answers
47 views

A seemingly basic PEMDAS problem… [duplicate]

There's one of those meme-type images posted on Facebook with the equation 6/2(1+2), challenging you to solve it. So, parenthesis first, ...
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3answers
21 views

Why variables in directly proportinality are multipiled

Why variables (RHS) in directly proportionality are always multiplied. Suppose the Newton's 2nd law $$F \propto m$$ $$F \propto a$$ $$F \propto m*a$$ Please don't give a rigorous proof. I just want ...
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2answers
54 views

Real Analysis; injective and surjective functions

Let $f$ and $g$ functions from $\mathbb{R}$ to $\mathbb{R}$ given by $$g(x)= x^2-x$$ and $$f(x)= -\sin x$$ i) Is $g$ injective? ii) is $g$ surjective? iii) is $g$ invertible? iv) is $f$ injective? ...
2
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1answer
61 views

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find minimum value of $t$ such that the predefined condition is satisfied for all natural numbers $x_i$'s

If $x_1^3+x_2^3+\ldots+x_t^3=2002^{2002}$, find the minimum value of $t$ such that the predefined condition is satisfied for all natural number $x_is$. My attempt: I took modulo $9$ on both sides ...
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3answers
83 views

How exactly does the response “infintely many” answer the question of “how many”?

I admit that the level of this question is roughly about middle school, but this is what the question asks: The ratio of nickels to dimes to quarters is 3:8:1. If all the coins were dimes, the ...
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2answers
55 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
2
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4answers
35 views

Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...
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2answers
29 views

Proving that function with domain (-1,1) is injective.

Function $g\colon (-1,1) \rightarrow \mathbb R$ is defined by $g(x)=\dfrac{x}{1-x^2}$. Prove that $g(x)$ is injective. Work: I shifted the equation so that it ends up like ...
4
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1answer
39 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. ...
2
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5answers
41 views

Calculate the height of a building

This question I really need help with, I simply do not know where to start! Anyone can help, all I can offer is supreme thanks. Please include method. I don't want simple answers which don't help me ...
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0answers
15 views

Substitution in multiple integrals, rewriting variables

I have this problem that seems almost laughably simple, but has stumped me for some time. For the variables $ u = x^2 - y^2 $ $ v = x$ $y$ Under the condition $ u > 0 $ We're simply asked to ...
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5answers
94 views

How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$

According to wolfram alpha this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$ But how do you show this? I know of no rules that works with addition inside square roots. I noticed I could do ...
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2answers
30 views

The zoo car park

In the zoo car park there were 780 vehicles. Most of the vehicles were cars,but one section contained coaches only.the ratio of cars to coaches was 11:2. How many vehilces were cars and how many were ...
2
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2answers
34 views

Question about $I(n)=x^n \cos x$

If $I(n)=x^n \cos x$, find $I(n)$ in terms of $I(n-2)$. This is a reduction formula question.
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2answers
56 views

Logarithm of a negative number

We know this identity: $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$ Suppose both $a$ and $b$ are negative. Then the left-hand size evaluates to something, it is a logarithm of a positive number (minus signs ...
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3answers
35 views

inverse of a function f(x)..change x and y

Find the inverse of the function $f(x)= (2x-1)/(x^2-1).$ we switch the x and y letters and then solve the the equation...but it became kind of complicated while solving
2
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0answers
46 views

Don't know when to add negative numbers

I'm definitely not a math person and only did general mathematics in high school, and unfortunately, not paying as much attention to that as I should have. Well, I'm doing Discrete Mathematics in my ...
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10answers
2k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
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1answer
16 views

How to handle a double inequality where all 3 spots have unknowns

Given the problem $4x \lt 2x + 1 \le 3x + 2$ solve for x. I'm not sure how to go about solving this problem. No matter how I subtract or add the x's or multiply/divide the coefficients I cannot ...
4
votes
1answer
37 views

Need help solving exponential equation $2\mathrm{e}^x=5-\mathrm{e}^{-x}$

I need help solving $2\mathrm{e}^x=5-\mathrm{e}^{-x}$. I've tried many ways of solving it but I keep getting the wrong answer. By the way, my book says the solutions are $x=-1.518$ and $x=0.825$ ...
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1answer
8 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
2
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1answer
31 views

Correct equation for this question

The question is, A small hydroelectric generating station can produce 17 MWh of energy in 12 months. AFter 4 months of operation, another generator is added. This additional generator can produce 11 ...
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0answers
35 views

Find the distance between Manhattan and Brooklyn. [on hold]

Two travelers, Manuel and Brooke, set out from two places, Manhattan and Brooklyn, and at the same time. Manuel from Manhattan with a design to pass through Brooklyn, and Brooke from Brooklyn to ...
1
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1answer
28 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
0
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1answer
17 views

Velocity problem

Beni and Gal compete in running on straight track AB, that distance is 50m. They both started from point A on that track and run to point B. Beni leaped first. Gal leaped one second after Beni and ...
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1answer
22 views

System of equations - lagrange

Solve this system of eq: $$\begin{cases} yz &= 2 \lambda x &\,\,\,(a)\\ xz &= 2 \lambda y &\,\,\,(b)\\ xy &= 2 \lambda z &\,\,\,(c) \\ x^2+y^2+z^2−3&=0 &\,\,\,(d) ...
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1answer
21 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$