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-2 votes

length of hypotenuse where only part know

I would say the simplest way to work this out is but using scale factors. You can divide the section $L=35+x$, and now a scale factor is $15/8 = 1.875$. Thus, $(35+x)/1.875 = x$. Since scale factor is ...
Marx Carton's user avatar
0 votes

Showing $\frac{\frac1{\cos A}-\frac1{\sin A}}{\frac1{\cos A}+\frac1{\sin A}}=\frac{\frac{\sin A}{\cos A}-1}{\frac{\sin A}{\cos A}+1}$

@Ned answered this question as simply as possible; "Multiply top and bottom by sin A" I can't believe I didn't see that. Thank you Ned.
Ryan's user avatar
  • 291
2 votes

Find number of solutions to the equation $\sin(6\sin x)=\frac{x}{6}$.

If $x$ is a root of the equation, so is $-x$. Furthermore, $x=0$ is a root. For these reasons, we only enumerate the number of positive roots. Since $\frac{x}{6}>1$ for $x>6$, for every positive ...
Mostafa Ayaz's user avatar
  • 32.2k
0 votes
Accepted

What am I doing wrong, when calculating sequence limit $\left|\sin \left(\pi \cdot \sqrt[3]{n^3+n^2}\right)\right|$

For $b_n$ which satisfies $b_n \to 1$, $a_n$ be $$a_n = \left|\sin(\pi n b_n)\right|$$ Your attempt is Since $b_n \to 1$, $\pi nb_n$ may be approximated to $$\pi nb_n \approx \pi n$$ which is False. ...
bFur4list's user avatar
  • 2,651
2 votes
Accepted

Are $\sin{x}$ degrees and $\sin{x}$ radians considered different functions?

In math, it is almost always radians. In fact, we omit radians and treat $x\,\mathrm{rad}$ as a number $x$. So, $\sin(x)$ means $\sin(x\,\mathrm{rad})$. Degrees are rarely used and denoted using $^\...
ultralegend5385's user avatar
0 votes

Question regarding cuberoots of integral multiples of i

What's your main motive??? To do in the most simplest way possible $$z^3=8i \implies z^3+(2i)^3=0 $$ $$\implies (z+2i)(z^2-2zi-4)=0 $$ So either $z=-2i$, or by the quadratic formula, $z=\frac{2i±\sqrt{...
Gwen's user avatar
  • 1,075
0 votes

Question regarding cuberoots of integral multiples of i

If you write $z^{3}=8i$ as $z^{3}=(-2i)^{3}$ (since $i^3=-i$) you will see that the scale is by factor 2. So the point you are looking at is $(2\cos{\frac{\pi}{6}},2\sin{\frac{\pi}{6}})$ and the ...
Red Five's user avatar
  • 1,485
1 vote

Find value of $2A$ if $A=\frac{3\tan\left(A\right)}{1-\tan\left(A\right)}-1$

Let us start by inverting the homographic relationship : $$A=\frac{4 \tan A -1}{1- \tan A} \ \iff \ \tan A=\frac{A+1}{A+4}=1-\frac{3}{A+4}$$ The idea is to use the following expansion of the "...
Jean Marie's user avatar
0 votes

Finding angles in a convex quadrilateral, given only the lengths and angle between opposing sides

I am trying my best to leave around a simple answer. I'm applying basic law of cosines:- $$BD^2=AB^2+DA^2-2ABDA\cos \angle BAD$$ $$BD^2=BC^2+CD^2-2BCCD\cos \angle BCD$$ Equating these, we get :- $$BC^...
Gwen's user avatar
  • 1,075
0 votes

Finding angles in a convex quadrilateral, given only the lengths and angle between opposing sides

The image defines a quadrangle with corners A, B, C and D in clockwise order. The 4 known side lengths are $a=|AB|$, $b=|BC|$, $c=|CD|$ and $d=|DA|$. Denote the angles inside the quadrangle at the 4 ...
R. J. Mathar's user avatar
  • 2,519
1 vote

Find $\int_{0}^{\pi/2} \frac{\mathrm{d}\theta}{(\sin\theta+1)^2}$.

Use $u=x+\sqrt{1+x^2} $(for the original integral, this substitution is also not a bad choice), then $$x=\frac{1}{2}(u-1/u),\quad {\rm d}x=\frac12(1+1/u^2){\rm d}u.$$ So $$\int_{0}^\infty \frac{{\rm d}...
Riemann's user avatar
  • 7,255
3 votes
Accepted

How to get the size of the angle bisector of a triangle using the law of cosines

Thanks for showing your work. The angle bisector theorem gives us $AC \times BD = AB \times CD$ like you mentioned. Notice that your cosine rule equations can be written as (EG Clearing denominators ...
Calvin Lin's user avatar
  • 69.2k
2 votes

How to get the size of the angle bisector of a triangle using the law of cosines

As you said $$\frac{BD}{DC}=\frac{AB}{AC} \implies\frac{BD^2}{AB^2}=\frac{DC^2}{AC^2}...(i)$$ By law of cosines, $$BD^2=AD^2+AB^2-2ADAB\cos \frac{\angle A}{2}$$ $$DC^2=AD^2+AC^2-2ADAC\cos \frac{\angle ...
Gwen's user avatar
  • 1,075
1 vote

Is it true that $\sin^2((2n+1) \pi x) \geq c$ occurs often (in precise sense)?

Context Your question is very similar to this one: Sine function dense in $[-1,1]$, about the density of the sequence $sin(n)$. The idea of the proof is to prove that $\mathbb{Z}+\pi \mathbb{Z}$ is ...
Jean's user avatar
  • 36
1 vote

Does $ \frac{\sin(\theta-\alpha)}{\sin\alpha}=\frac{\cos(\theta+\gamma-\alpha)}{\cos(\gamma-\alpha)}$ have an analytical solution for $\alpha$?

You can rewrite your Equation (1) as $$\sin\!\left(A-B\right)\cos\!\left(A+B\right)=-\sin\!\left(A-C\right)\cos\!\left(A+C\right)$$ with $$A=\frac12\left(\theta+\gamma-2\alpha\right)\,,\quad B=\frac12\...
Toffomat's user avatar
  • 2,255
4 votes
Accepted

Does $ \frac{\sin(\theta-\alpha)}{\sin\alpha}=\frac{\cos(\theta+\gamma-\alpha)}{\cos(\gamma-\alpha)}$ have an analytical solution for $\alpha$?

Short answer: $$\alpha=\arctan\left(\frac1{\cot\theta-\tan\gamma+\sqrt{\tan^2\gamma+\cot^2\theta+1}}\right)$$ is the positive $\alpha.$ Directly from the geometric question, if we write in coordinate ...
Thomas Andrews's user avatar
1 vote

Circles,projectile motion and parabola

The first thing I would investigate would be to look at circles through $O$ and $R$ that also intersect the arc of the parabola between $O$ and $R$. If $P$ is one of those points of intersection then $...
David K's user avatar
  • 98.5k
0 votes

The fastest solution for $\int \sqrt{x^2+1}\,dx$

$\frac {e^{2\sin^{-1} x} - e^{-2\sin^{-1} x}}{8}$ is ugly and it has a nice simplification. $\frac {e^{2\sin^{-1} x} - e^{-2\sin^{-1} x}}{8} = \frac {x\sqrt {x^2+1}}{2}$ You might notice that if we ...
user317176's user avatar
  • 11.2k
4 votes

The fastest solution for $\int \sqrt{x^2+1}\,dx$

Another possible approach based on the integration by parts method: \begin{align*} \int\sqrt{x^{2} + 1}\mathrm{d}x & = x\sqrt{x^{2} + 1} - \int\frac{x^{2}}{\sqrt{x^{2} + 1}}\mathrm{d}x\\\\ & = ...
Átila Correia's user avatar
0 votes

How moving the center the of the Unit Circle affects the cosine function

The circle is an entity comprising its center and circumference. We are considering parametric equation which is a combined relation for the circle centered at the origin: $$ x= \cos \theta \text { ...
Narasimham's user avatar
  • 40.5k
15 votes
Accepted

Why the sine wave of a sine wave returns a sort of cubic sine wave?

The $n$-th iterate of sine (denoted $\sin^{(n)}$) has been studied in great detail by N.G. de Bruijn, Asymptotic Methods in Analysis, pages 157–166. He finds, for $0<x<\pi$, that $$\sin^{(n)}x=\...
Carlo Beenakker's user avatar
2 votes

Trigonometry inequality for sine of 5 angles

For the left inequality. If $x_1 + x_2, x_2 + x_3, x_3 + x_4, x_4 + x_5, x_5 + x_1 \le \pi$, clearly the inequality is true. In the following, WLOG, assume that $x_1 + x_2 > \pi$. We have \begin{...
River Li's user avatar
  • 37.8k
1 vote
Accepted

Find value of $2A$ if $A=\frac{3\tan\left(A\right)}{1-\tan\left(A\right)}-1$

To get rid of the discontinuities, multiply everything by $(1-\tan(A))\cos(A)$ which makes that we look for the zero's of function $$f(A)=(A+4) \sin (A)-(A+1) \cos (A)$$ As said in comments, the ...
Claude Leibovici's user avatar
2 votes

Finding $A+B+C$ in $\cos^4(x)=A+B\cos(2x)+C\cos(4x)$?

If $$\cos^4(x)=A+B\cos(2x)+C\cos(4x)$$ is an identity, it must be true for small values of $x$. So, using Taylor $$1-2 x^2+\frac{5 x^4}{3}+O\left(x^6\right)=(A+B+C)-2 (B+4 C)x^2+\frac{2}{3} (B+16 C)x^...
Claude Leibovici's user avatar
2 votes

Finding $A+B+C$ in $\cos^4(x)=A+B\cos(2x)+C\cos(4x)$?

$\cos^4 x = A + B\cos 2x + C\cos 4x\\ \cos 4x = 8\cos^4 x - 8\cos^2x + 1\\ \cos 2x = 2\cos^2 x - 1\\ u = \cos^2 x\\ u^2 = A+ 2Bu - B + 8Cu^2 - 8Cu + C\\ u^2 = (A-B+C)+ (2B-8C)u + 8Cu^2$ Setting the $u^...
user317176's user avatar
  • 11.2k
4 votes

Finding $A+B+C$ in $\cos^4(x)=A+B\cos(2x)+C\cos(4x)$?

Expand $\cos^4(x)$ using the double angle formula $\cos^2(\theta) = \frac{1}{2}(\cos(2\theta) + 1)$: \begin{align*} \cos^4(x) &= (\cos^2(x))^2,\\ &= \left(\frac12(\cos(2x)+1)\right)^2,\\ &=...
Quick_Fix's user avatar
3 votes
Accepted

Finding $A+B+C$ in $\cos^4(x)=A+B\cos(2x)+C\cos(4x)$?

There is no reason to use Vieta’s Formulas, I believe. Take the equation you found: $$(8C-1)\cos^4(x)+2(B-4C)\cos^2(x)+A-B+C =0$$ This must be true for all values of $x$. So, you can essentially treat ...
solasky's user avatar
  • 160
2 votes

Error in graphing the polar equation $r=-1+\cos(\theta)$. I get a different answer than the book for cos(30degrees). Am I wrong?

It's wrong. Here is why. You can directly put the value and check. It's wrong for $90°$ part as well. For a more general approach:- $f(x)=-1+\cosθ$ has maximum value of $0$ at $θ=0°$ and minimum ...
Gwen's user avatar
  • 1,075
1 vote
Accepted

Solving a Two-by-Two System of Trigonometric Equations

Hints: Step 1, subtract the two equation. You get a single equation that contain $\sin\theta$ and $\cos\theta$. Step 2, move $\sin\theta$ term to one side, $\cos\theta$ term to the other side. Check ...
Andrei's user avatar
  • 37.4k
2 votes

Maximum value of $\sin(x)-\sin^3(x)$ without using calculus

We apply Vieta's Formulas. Let $a$ be the maximum value and render $u=\sin x$. We see $a$ such that $-u^3+u-a$ will have a double root between $u=0$ and $1$. Call the double root $r_2$ and the ...
Oscar Lanzi's user avatar
  • 39.5k
1 vote

Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $

REMARKS.- $(1)$ You have to edit Machin's equality : which must be $$\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right) $$ (what you have written gives $\pi$ ...
Piquito's user avatar
  • 29.9k
3 votes
Accepted

Maximum value of $\sin(x)-\sin^3(x)$ without using calculus

Equivalently, let us look at $f(x) = x-x^3$ for $x\in [-1, 1]$. Clearly the maximum is for some $x\in (0, 1)$, as for other $x$ in the domain, $f(x)=x(1-x^2)\leqslant 0$. To use AM-GM, consider ...
Macavity's user avatar
  • 46.4k
0 votes

Maximum value of $\sin(x)-\sin^3(x)$ without using calculus

This is equivalent to finding the local extrema of the cubic function $f(u)=u-u^{3}$ under the substitution $u=\sin x$. Observe that if a cubic function attains a local extremum $m$ then $-u^3+u-m$ ...
Jam's user avatar
  • 10.3k
-1 votes

Prove that $x-a \sin(x)=b$ has one real solution, where $0\lt a \lt 1 $

By Banach fixed-point theorem : Let $g(x) = a \sin (x) + b$. Then, $$ g(x) = x \iff x - a \sin(x) = b. $$ Therefore, it is sufficient to prove that $g$ has a unique fixed point. However, $$ g'(x) = a\...
Sarvesh Ravichandran Iyer's user avatar
4 votes

How to find an acute angle of a right triangle inscribed in a square?

I'm afraid you're likely to kick yourself…  (This doesn't need any trigonometry, just simple angle sums.) First, consider the angles in the bottom triangle.  Its left-hand angle is marked as $\alpha$, ...
gidds's user avatar
  • 179
6 votes
Accepted

How to find an acute angle of a right triangle inscribed in a square?

$?+90+(90-\alpha)=180$ so $?=\alpha$.
marty cohen's user avatar
0 votes

Graphing a function which is a product or sum of 2 functions?

We have a fairly quick way of doing the sum of two functions. The same is not true for the product, although you can get a bit of intuition into graphing a product. The quickest way to graph a sum is ...
Nicholas Smith's user avatar
1 vote

Range of the function $ f(x) = \frac{1}{1 - 2 \cos(x)}$

None of the answers are correct. Your math is. You can approach the problem "backwards". $\exists c| f(c)=2 \implies $ A,B, and D are false. $\frac{1}{1-2\cos (c)}=2\implies 1/2=1-2\cos (c)\...
TurlocTheRed's user avatar
  • 5,688
4 votes

Is it true that $\sin^2((2n+1) \pi x) \geq c$ occurs often (in precise sense)?

The claim is true. Indeed, consider the sequence $(z_n)_{n\in\mathbb N}$ of points of the unit circle $$\mathbb T=\{(u,v)\in\mathbb R^2:u^2+v^2=1\}$$ such that $z_n=(\cos (2n+1)\pi x,\sin (2n+1)\pi x)$...
Alex Ravsky's user avatar
1 vote

General solution of $\cos(3\theta) - \cos(\theta) = 0$

Plugging in the candidate solution angles into the original equation, we see that any integer multiple of $\frac{\pi}{2}$ (90°) will satisfy it. Your two solutions are actually the same. Method 1 ...
Dan's user avatar
  • 15.2k
3 votes

General solution of $\cos(3\theta) - \cos(\theta) = 0$

You first solution set $\{\frac{(2n+1)\pi}{2},n\pi\}$ is equal to the second $\{\frac{n\pi}{2},n\pi\}$.
Shean's user avatar
  • 917
1 vote

In triangle ABC, $\sin A + \sin B + \sin C ≤ 1$. Prove that $\min\{A + B, B + C, C + A\} < 30°$.

WLOG, assume $A \leq B \leq C$, then $0 < A \leq 60^\circ \leq C.$ Suppose $C \leq 90^\circ$, then $B \geq (180^\circ - C)/2 \geq 45^\circ$. Therefore $$\sin A + \sin B + \sin C \geq \frac{1}{\sqrt{...
ioveri's user avatar
  • 1,441
1 vote

Using Poisson Kernel Identity

For simplicity, I suppose $\omega = 1$ which obviously does not reduce the generality of what follows. The denominator of the left hand side $L$ of (1) is equal to $(1-pe^{it})(1-pe^{-it})$. The idea ...
Ulysse Keller's user avatar
1 vote

In triangle ABC, $\sin A + \sin B + \sin C ≤ 1$. Prove that $\min\{A + B, B + C, C + A\} < 30°$.

WLOG, assume that $a \leq b \leq c$, so that $A\leq B \leq C$ and $\min\{A+B,A+C,B+C\}=A+B$. Next, if $30^{\circ}<B<90^{\circ}$, $\sin{B}>\frac{1}{2}$, forcing $\sin{C}<\frac{1}{2}$. This ...
mathy_mathema's user avatar
3 votes
Accepted

$\cos 2θ = 2\cos^2θ -1 = \cos^2θ + \sin^2θ$?

Your working suggests that $$\cos^2 \theta-1=\sin^2 \theta$$ which can be rewritten as $$\cos^2 \theta \color{red}- \sin^2 \theta = 1$$ It is not true, it should be $$\cos^2 \theta + \sin^2 \theta = 1$...
Siong Thye Goh's user avatar
1 vote
Accepted

Find the center point of ellipse by only 2 points on it and radiuses

Given two points $(x_1,y_1),(x_2,y_2)$ on the ellipses and semi-axes lengths $a, b$ for axes that are axis aligned, the equations for $h, k,$ where $(h,k)$ is a center, in M2 are ...
Jan-Magnus Økland's user avatar
2 votes
Accepted

Can we show that the following integral is independent of $\phi_0$ without computing the integral itself?

Not very elegant: by using power reduction formulae and product to sum formulae, the integrating function can be written \begin{align} \sin^4\left(\dfrac{\phi}{2N}\right)\sin^2\left(\phi+\phi_0\right) ...
Vincenzo Tibullo's user avatar
1 vote

Is $\cos(\alpha x + \cos(x))$ periodic?

$\newcommand{\boxedtext}[1]{\require{enclose}\enclose{box}{\textbf{#1.}}~~}$ Define $f_\alpha(x) = \cos(\alpha x + \cos(x))$. Below, I prove the following. Claim: The function $f_\alpha$ satisfies ...
Drew Brady's user avatar
  • 3,690
2 votes
Accepted

Solve for $x$ in $y=x\operatorname{arccot}(x)$

Hint You can have good approximations of the inverse of $$y=x \cot ^{-1}(x)$$ straing from the Taylor series of the arctangent function $$\cot ^{-1}(x)=\frac \pi 2-\sum_{n=0}^\infty \frac{(-1)^{ n}} {...
Claude Leibovici's user avatar

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