Why does square root of 50 resembles sin(45°)? I was formulating the steps to calculate the area of a fence plank, and was looking at the sloped shape of the top of a plank (beeing a right angle tip).
My example plank had a width of $10$ cm, and the side of the sloped tip got the length
$\sqrt{50}$.
That is $7.071...$
After that I looked at $\sin(45°)$ which is
$0.7071...$
(Even if this story might seem unclear, the numbers are clearly similar)
Is there a connection between these numbers?
Is there a geometric explanation why these numbers are similar (except for a step of the comma)?
The longer versions are
$7.07106781186$
$0.70710678118$
 A: The exact value of $\sin(45^{\circ})$ is $\frac{\sqrt 2}2$.
On the other hand, $\sqrt{50} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$.
You can see from these two expressions that $\sqrt{50}$ is exactly 10 times as large as $\sin(45^{\circ})$.
Now, if you want to know why $\sin(45^{\circ}) = \frac{\sqrt 2}2$, draw an isosceles right triangle with legs of length $\sqrt{2}$, and use the Pythagorean Theorem to find the length of the hypotenuse.
A: Draw a right isosceles triangle, making the sides next to the right angle $\sqrt{50}$ units long. The other two angles will be $45^\circ$ and the longest side (the hypotenuse) will be, as per Pythagoras' theorem:
$$\sqrt{(\sqrt{50})^2+(\sqrt{50})^2}=\sqrt{50+50}=\sqrt{100}=10$$
and now you can see on this triangle that
$$\sin 45^\circ=\frac{\text{opposite side}}{\text{hypotenuse}}=\frac{\sqrt{50}}{10}$$
which explains why $\sin 45^\circ$ and $\sqrt{50}$ have the same digits. (The latter is just $10$ times the former!).
A: $\sqrt{50}=\sqrt{2*25}=5\sqrt{2}$
It is known that $\sin 45^o$ is $\frac{\sqrt{2}}{2}$ So there's your answer.
