Length of A Diagonal Line of Square

Question

After watching this video to calculate the length of diagonal of square , a question arises to me is :

Why is length of diagonal of square $$\frac{\text{side}}{cos45^0}$$?

Why $cos45^0$ ?If i write $sin45^0$ instead of $cos45^0$ , will it be wrong?

Answer 1

In a square, diagonal (let's say $d$) and any two sides (let's call them $a$) form a right angled isosceles triangle. In that triangle $\dfrac{a}{d}=\sin45^{\circ}=\cos45^\circ.$

Because, $\sin{\theta}=\dfrac{opposite}{hypotenuse}=\dfrac{adjacent}{hypotenuse}=\cos{\theta}$, where $\theta$ is the acute angle and in this case it is $45^{\circ}$,$45^{\circ}$ and also $opposite= adjacent=a$.

So, $d=a\sqrt{2}$

So, $\sin45^{\circ}$ is also right.