Question 820967
Maybe you meant {{{cos(pi/4)}}} .
The whole circle, {{{360^o}}} is {{{2pi}}} measured in radians,
so {{{pi/2}}} is {{{90^o}}} , a right angle.
A square has 4 congruent sides (equal length) and 4 right angles.
A diagonal divides a square into two isosceles right triangles,
with a right angle ( {{{90^o}}} otr {{{oi/2}}} ,
and two congruent (equal measure) angles,
measuring {{{45^o}}} or {{{pi/4}}} .
The trigonometric functions of such an angle can be calculated as the ratios of the sides of such a triangle.
If I make the length of the sides of my square {{{1}}} unit (of whatever unit you want), how long is that diagonal
{{{drawing(300,300,-0.1,1.1,-0.1,1.1,
rectangle(0,0,1,1),rectangle(0.95,0,1,0.05),
green(line(0,0,1,1)),
locate(0.47,0,1),locate(1.01,0.53,1)
)}}} The diagonal is the hypotenuse of a right triangle with legs measuring {{{1}}} .
The length of the diagonal is {{{sqrt(1^2+1^2)=sqrt(1+1)=sqrt(2)}}} .
The sine of one of those {{{pi/4}}} angles is the ratio of opposite leg, divided by hypotenuse, so
{{{sin(pi/4)=1/sqrt(2)}}}
However, we do not like to see square roots in denominators, so we multiply numerator and denominator times {{{sqrt(2)}}}.
{{{sin(pi/4)=sqrt(2)/(sqrt(2)*sqrt(2))}}}
{{{sin(pi/4)=sqrt(2)/2}}}