10
votes
Accepted
Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$
We have that for $\theta\in(0,\pi/2)$, $f(\theta)=\sec \theta -\cos \theta$ is an increasing function and
$$\sec \theta-\cos \theta =1 \implies \cos \theta = \frac{\sqrt 5-1}2 =\frac 1\varphi$$
that ...
- 142k
7
votes
Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$
For angles less than 60 degrees, $\cos(3x)$ is a decreasing function of $\cos(x)$. So apply triple angle identity twice to $\cos(52)<(\sqrt{5}-1)/2$ to get the equivalent formulation $\cos(72) > ...
- 19k
4
votes
Is there an $x$ such that $\cos(x)=\sin(x)=0$?
Suppose there was a point $x\in\mathbb{R}$ such that
$$\cos x=\sin x=0.$$
Then we would have
$$0=\cos^2x+\sin^2x=1,$$
which is a contradiction. Thus such a point cannot exist.
- 4,553
4
votes
Accepted
When can the value of a given number be obtained as the absolute value of a sum of complex roots of unity?
One observation is that the absolute value of a sum of roots of unity is an algebraic integer. If $m$ and $\mu$ are integers, $\rho(m,\mu)$ is an algebraic number, but if is not an algebraic integer (...
- 420k
4
votes
using trig sub vs u-sub with $ \int_0^1 x^3 \sqrt{1 - x^2} \, dx $?
This is a good exercise because the answer is do both!
The "Pythagorean" expression (a square-root of a sum or difference of squares) suggests a trigonometric substitution. With $x = \sin t$,...
- 18.6k
3
votes
Accepted
Prove that $\sin^n (2x) + (\sin^n x - \cos^n x)^2 \leq 1$
Assume that $n$ is a positive integer.
Let $a = \sin x$ and $b = \cos x$.
Then $a^2 + b^2 = 1$ and $|2ab| \le a^2 + b^2 = 1$.
Let $$f(n) := (2^n-2) a^n b^n + a^{2n} + b^{2n}.$$
We have $f(1) = a^2 + b^...
- 25.9k
3
votes
Accepted
Relationship between e and sine
We shall use that for all $s\in\mathbb R$,
$$|1-e^{is}|=|e^{is}-1|=|e^{is/2}(e^{is/2}-e^{-is/2})|=|2ie^{is/2}\sin(s/2)|=|2\sin(s/2)|.$$
Let $t=ka\sin\theta$.
$$\begin{align}I(Q)&=b^2\left|\sum_{n=...
- 2,855
3
votes
Compute the area of Quadrilateral $ABCD$
This will be my approach to this problem. I shall add a brief explanation as well:
So this is how I go about it:
Please note that I forgot to mark the point of intersection of the two diagonals, ...
- 237
3
votes
Accepted
Do we have $\prod_{n=1}^{\infty}\left(1+\frac{\tan\left(n\right)+1}{n\left(\tan\left(n+1\right)+1\right)}\right)=^?0$
Because $n$ is a natural integer versus irrational $\pi$, then $n+k\pi$ for all $n$ has a uniform distribution over $[-\pi/2,\pi/2]$ for some $k\in \mathbb Z$.
So, $\tan(n)$ values between $[0,1]$ for ...
- 359
2
votes
How does one come up with the continued fraction for the arctangent? $\arctan x=\frac{x}{1+}\frac{x^2}{3+}\frac{(2x)^2}{5+}\cdots$
For reference, here is a computation using generating functions. We show that $$\cfrac{z}{1+\cfrac{z^2}{3+\cfrac{(2z)^2}{5+\cfrac{(3z)^2}{7+\dots}}}}$$ exists and equals $\arctan z$ for $z\in\mathbb{C}...
- 33.6k
2
votes
How to express this $\sin^8{\frac{2\pi}{7}}\sin^7{\frac{3\pi}{7}}$ in terms of $\sin{\frac{\pi}{7}}$,$\sin{\frac{2\pi}{7}}$,$\sin{\frac{3\pi}{7}}$
I'm not sure expanding it out is a terrible way to go actually, especially with some shortcuts. Here, hold my root beer.
Let $x = \frac{\pi}{7}$ just so we don't have to type that over and over. The ...
- 2,101
2
votes
How to express this $\sin^8{\frac{2\pi}{7}}\sin^7{\frac{3\pi}{7}}$ in terms of $\sin{\frac{\pi}{7}}$,$\sin{\frac{2\pi}{7}}$,$\sin{\frac{3\pi}{7}}$
The usual boaring but useful technique (using Euler's formulas and the binomial coefficient theorem) gives you $\sin^8(2x)\sin^7(3x)=$
$$\frac1{2^{14}}\left(\begin{align}&1365\sin x+2226\sin(3x)+...
- 2,855
2
votes
Accepted
Find two ratios given an inequality about a triangle with integer coordinates
This problem is, in fact, about inequalities and the discreteness of integers besides the basic knowledge of geometry.
$$\begin{aligned}&\quad\quad1\\
&>(|AB|+|BC|)^2-8\cdot\text{Area}(\...
- 3,328
2
votes
Can someone explain the graph of sin(x^2+y^2) = x?
Is the question $sin(x^2+y^2)=0.5$ ?
For equation $sin\theta=0.5$, there is infinite solutions.
In form of $\theta=n\pi+(-1)^n\alpha$ where $\alpha=\pi/6$
Then for each positive solution $\theta$, you ...
- 824
2
votes
Accepted
Quaternion slerp trig for solving scalars of vectors
I have added two purple line segments to your figure, each perpendicular to the line labeled $v_1$.
Since the line labeled $v_1$ is parallel to the line labeled $k_1v_1,$ the two new line segments are ...
- 87.5k
2
votes
Without any software and approximations prove that $\sec(52^{\circ})-\cos(52^{\circ})>1$
Update:
Remarks: We can calculate $\cos (3 \cdot 52^\circ)$ in radical form. Then use $\cos 3u = 4\cos^3 u - 3\cos u$ to prove $\cos 52^\circ < \frac{\sqrt 5 - 1}{2}$. This is based on @eyeballfrog'...
- 25.9k
2
votes
Compute the area of Quadrilateral $ABCD$
Alternative approach:
Area$(\triangle ADC) = \dfrac{1}{2} \times \overline{AD} \times [ ~\overline{AC} \sin(105^\circ)].$
Area$(\triangle ABC) = \dfrac{1}{2} \times \overline{BC} \times [ ~\overline{...
- 25.7k
2
votes
Compute the area of Quadrilateral $ABCD$
Let the intersection of the two diagonals be $O$.
And let $ AO = x , OC = y, AD = z $
Then it follows from the law of sines that
$ DO = a x \hspace{25pt}$ where $ a = \dfrac{\sin(105^\circ)}{\sin(30^\...
- 11.1k
2
votes
Find $X$ from convex quadrilateral $ABCD$
So, this'll be my own approach to this problem. I'll add a brief explanation as well:
This is how I go about this:
1.) Locate the circumcenter of $\triangle BDC$ at point $O$ (Note: Yes the ...
- 237
2
votes
Accepted
Find $X$ from convex quadrilateral $ABCD$
Here is a solution using complex numbers, with the hope you have already studied them.
I will not make any difference between a complex number and the point associated with it in the so-called Argand ...
- 67k
2
votes
Find Angle $\alpha$ from the triangle
So this'll be my approach. I'm going to add a brief explanation too.
Here's how I did it:
1.) Mark the $\triangle ABC$ with all the appropriate angles and mark the midpoint of segment $AC$ as $D$.
2.)...
- 237
2
votes
Proving $\frac{\cos x}{\sin x} = \left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} +1\right)$
Hint:
For the first item in parenthesis, use the identity $\tan x = \displaystyle \frac {\sin x}{\cos x}$ and put under a common denominator.
For the second item in parenthesis, put under a common ...
- 3,048
2
votes
Proving $\frac{\cos x}{\sin x} = \left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} +1\right)$
$$
\left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} +1\right) = \left( \frac{1}{\cos x} - \frac{\sin x}{\cos x} \right) \left( \frac{1}{\sin x} +1\right)\\
=\left( \frac{1- \sin x}{\...
- 21
2
votes
Find exact value of $\tan (\frac{\pi}{12})$ given that $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
$$\frac{1}{2}=\sin \frac{\pi}{6}=\sin (2 \times \frac{\pi}{12})=2\times\sin \frac{\pi}{12} \times \cos \frac{\pi}{12}=2\times \frac{\sqrt3 -1}{2\sqrt{2}}\times \cos \frac{\pi}{12} $$
$$\cos \frac{\pi}{...
- 5,161
2
votes
Find exact value of $\tan (\frac{\pi}{12})$ given that $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
As per the suggestion of Santos:
$$\sin(\pi/12)=\frac{\sqrt{3}-1}{2\sqrt{2}}\implies \tan(\pi/12)=\frac{\sqrt{3}-1}{\sqrt{(2\sqrt{2})^2-(\sqrt{3}-1)^2}}=\frac{\sqrt{3}-1}{\sqrt{4+2\sqrt{3}}}$$ Next ...
- 431
2
votes
Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
Another way would be to use the double angle formula
$$\sin 2\theta=2\sin\theta\cos\theta$$
With $\theta=\frac {\pi}{12}$, then
$$\sin 2\theta=\sin\frac {\pi}6=\frac 12$$
Thus
\begin{align*}
\cos\frac ...
- 5,382
2
votes
Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
Let $$\sqrt{2+\sqrt{3}}=\sqrt{x}+\sqrt{y}\implies x+y=2. xy=3/4 \implies x=3/2, y=1/2.$$
So $$\cos(\pi/12)=\frac{\sqrt{3}+1}{2\sqrt{2}}$$
OP is right.
- 431
2
votes
Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
Nothing wrong.
If you prefer, we can do some simplification.
Let me focus on $\sqrt{2+\sqrt3}$.
Let $$\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3}=x$$
$$2+\sqrt3+2-\sqrt3+2=x^2$$
Hence, we have $$\sqrt{2+\sqrt3}+...
- 145k
2
votes
Accepted
A question on a property of definite integrals that $\int_a^b f(x)dx =\int_a^b f(a+b-x)dx$
Let $$I=\int_\frac{\pi}{6}^\frac{\pi}{3}\frac{1}{1+\sqrt{(\tan(x)}}dx$$
Using the substitution $t=({\frac{\pi}{3} + \frac{\pi}{6}})-x=\frac{\pi}{2}-x\implies dt=-dx$
$$I=\int_\frac{\pi}{6}^\frac{\pi}{...
- 422
1
vote
Stable set for $f(x) = \frac{\pi}{2}\sin(x)$
Assuming your angles are in $(-\pi, \pi]$, there are two stable points: $\pm \pi/2$ (and one unstable point: $0$). This can be seen by plotting $y=x$ and $y=f(x)=\frac{\pi}{2}\sin x$ on the same graph....
- 2,349
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