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.

learn more… | top users | synonyms (1)

1
vote
1answer
14 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.
0
votes
4answers
29 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?
2
votes
1answer
43 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 ...
0
votes
3answers
50 views

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$. Do I use the distance formula with the variable or not?
-1
votes
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
vote
3answers
77 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 ...
1
vote
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.
0
votes
2answers
18 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. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
0
votes
4answers
60 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 ...
0
votes
0answers
40 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 ...
1
vote
0answers
29 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 ...
0
votes
0answers
14 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 ...
2
votes
5answers
39 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 ...
0
votes
3answers
38 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 ...
0
votes
2answers
1k views

The distance the Earth travels in one day

Find the distance that the Earth travels in one day in its path around the Sun. Assume that a year has 365 days and that the path of the Earth around the Sun is a circle of radius 93 million miles. ...
1
vote
3answers
20 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 ...
0
votes
1answer
11 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 ...
2
votes
4answers
31 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? ...
0
votes
2answers
39 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.
-1
votes
1answer
31 views

question about rational expressions

i can't understand how to do this 5 question please help me ! Thank you
1
vote
1answer
38 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.
0
votes
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?
0
votes
4answers
44 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, ...
0
votes
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?
4
votes
3answers
246 views

Is $(0,0)$ a solution to $x^y-y^x=0$?

I am trying to determine if $(0,0)$ a solution to $x^y-y^x=0$. My hunch is that it is undefined since $0^0$ is an indeterminate form. To attempt to prove this, I have tried the usual "different ...
0
votes
2answers
50 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
votes
1answer
59 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 ...
4
votes
2answers
88 views

Determine whether this d is a metric?

Suppose we have a function $d:\mathbb{Z}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{R}$ where $\forall(x_{0},y_{0}),(x_{1},y_{1})\in \mathbb{Z}^{2}$, we have.. $d\left( \left( x_{0},y_{0} \right), ...
1
vote
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.
9
votes
10answers
1k 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)
1
vote
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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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
0
votes
2answers
60 views

Can you only take positive numbers into a square root?

I am thinking about when you are simplifying something like this $x(1/x)^{0.5}=x^{0.5}$ but if $x$ was negative then you have lost the sign why is this still valid or is it not?? Another example is ...
3
votes
5answers
93 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 ...
6
votes
3answers
160 views

An Inequality Problem with not nice conditions

How to show that $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac32$, where $a^2+b^2+c^2=3$, and $a,b,c > 0$ ?
0
votes
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 ...
2
votes
2answers
33 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.
0
votes
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) ...
1
vote
3answers
84 views

What does “$f(x,y)$ is strictly increasing in each argument” imply?

Say we have a function $f(x,y)$. below are what we know about $f(x,y)$ strictly increasing in each argument. $x$ and $y$ are natural numbers only, i.e., $0, 1, 2, ...$ Now we have a fixed number ...
0
votes
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 ...
-2
votes
1answer
27 views

a question on time, distance and speed

Ram and Gopi run towards each other starting from K and L respectively with respective speeds of 2 kmph and 3 kmph. After meeting each other, to reach L, if Ram takes 7 hours less than the square of ...
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 $x=0.825$ ...
3
votes
2answers
52 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
1
vote
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.
0
votes
1answer
20 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 $(2+6i)^6$
2
votes
1answer
199 views

Goat tethered in a circular field

I need help on this question. I was thinking of counting the area of the whole circle and then subtracting it with the area that is not eaten by the goat. But I don't know how to find this particular ...
1
vote
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 ...