MHB Can the Sum of Two Trigonometric Functions Be Less than pi/2?

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Prove that $\cos(\sin x))+\cos(\cos x))<\dfrac{\pi}{2}$.
 
Mathematics news on Phys.org
[sp]There's a minimum of calculator work here. I'm assuming that's okay since I don't know an exact expression for cos(1).

Let [math]f(x) = cos( sin(x) ) + cos( cos(x) )[/math] (Let [math]0 \leq x < 2 \pi[/math] ).

Then [math]f'(x) = sin(sin(x)) \cdot ( 1 + cos( cos(x) ) )[/math] will give relative maxs and mins.

The maxima are found to be at [math]x = 0, ~ \pi[/math]. We can easily show that [math]f(0) = f( \pi ) = 1 + cos(1) \approx 1.5032 < \dfrac{ \pi }{2}[/math]
[/sp]

-Dan
 
Let $y=\sin x$ and $z=\cos x$.
Then we want to know if the maximum value of $\cos(\cos x)+\cos(\sin x)=\cos y+\cos z$ is less than $\frac\pi 2$ with the boundary condition $y^2+z^2=1$.

We can find the extrema with Lagrange multipliers.
Define $\Lambda(y,z,\lambda)=\cos y+\cos z-\lambda(y^2+z^2-1). \tag 1$

The extrema occur for $(y,z)$ with:
$$\begin{cases}\Lambda_y=-\sin y -2\lambda y = 0\\
\Lambda_z=-\sin z -2\lambda z = 0 \\
\Lambda_\lambda = -(y^2+z^2-1) = 0 \end{cases} \implies
\begin{cases}-2\lambda = \frac{\sin y}{y} \lor y=0 \\
-2\lambda = \frac{\sin z}{z} \lor z = 0 \\
y^2+z^2=1\end{cases} \implies
\begin{cases}\frac{\sin y}{y} = \frac{\sin z}{z} \lor y=0 \lor z=0 \\
y^2+z^2=1\end{cases} \tag 2
$$
So $(y,z)$ is on the unit circle and $-1\le y,z \le +1$.
Note that $\frac{\sin t}{t}$ is an even function that is bijective on the interval $(0,1]$. Therefore $\frac{\sin y}{y} = \frac{\sin z}{z} \implies y=\pm z$.
It follows that the extrema are when $(y,z)$ is $\left(\pm \frac 1{\sqrt 2},\pm \frac 1{\sqrt 2}\right)$ and/or $(0,\,\pm 1)$ and/or $(\pm 1,\,0)$.

Thus: $\cos(\sin x)+\cos(\cos x) \le \max\left[ \cos\left(\pm \frac 1{\sqrt 2}\right) + \cos\left(\pm \frac 1{\sqrt 2}\right),\,\cos(0) + \cos(\pm 1)\right]
=\max\left[2\cos\left(\frac 1{\sqrt 2}\right),\, 1+\cos 1\right] \tag 3$

From the Taylor expansion we get that $\cos x \le 1-\frac {x^2}2+\frac{x^4}{4!}$.
So $2\cos\left(\frac 1{\sqrt 2}\right) \le 2\left(1-\frac {1/2}2 + \frac {(1/2)^2}{24}\right)=1.5+\frac 1{48} \le 1.5 + \frac 1{20} = 1.55$ and $1+\cos 1 \le 1+ \left(1-\frac 12 + \frac 1{24}\right) \le 1.5 + \frac 1{20}=1.55$.

Finally we have that $\frac\pi 2 > \frac{3.14}{2} = 1.57$, which completes the proof since $1.55 < 1.57$.
 
Last edited:
Using $\cos x \leqslant 1 - \frac{x^2}2 + \frac{x^4}{24}$ (as in Klaas van Aarsen's solution), $$\begin{aligned}\cos(\sin x) + \cos(\cos x) &\leqslant 1 - \frac{\sin^2x}2 + \frac{\sin^4x}{24} + 1 - \frac{\cos^2x}2 + \frac{\cos^4x}{24} \\ &= 2 - \frac{\sin^2x + \cos^2x}2 + \frac{\sin^4x + \cos^4x}{24} \\ &< \frac32 + \frac{(\sin^2x + \cos^2x)^2}{24} = \frac{37}{24} < 1.55 < 1.57 <\frac{\pi}2.\end{aligned}$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
28
Views
3K
Replies
7
Views
2K
Replies
7
Views
1K
Replies
2
Views
1K
Replies
5
Views
2K
Replies
2
Views
991
Back
Top