As far as I know sine has values between 1 and -1 therefore I don't know how to calculate this kind of questions with calculator! Could you help me please...
My personal favorite is to do it by the unit circle, File:Unit circle.svg - Wikipedia, the free encyclopedia
In this particular case, the angle t = 45 degrees, using the notation in the picture.
Now, we are going to use two basic facts about triangles, the Pythagorean theorem as well as the basic definition of the sine-function. First off, the sum of the angles in a triangle is 180. Meaning that if you know one angle is 90 and one is 45, you know that the third is 180-90-45 = 45 degrees
Having two equal angles, we know that the two sides not drawn in the picture are the same length. And since we're working in a unit circle, we know that the side drawn, the hypotenuse of the triangle, is 1. Thus, if we call the sides S we know by the Pythagorean theorem that
S^2 + S^2 = 1
or
S = 1/sqrt(2) [sqrt(2) means the square root of two!]
Now, all we need to do is take the basic definition of the sine-function in a right triangle, that is
sin(t) = opposite/hypotenuse
In our case we have sin(45) = (1/sqrt(2))/1 = 1/sqrt(2)
Hope that helps a little and wasn't just all confusing. My recommendation would be to step back and review the stuff I mentioned in the second paragraph; if you have that down this will be a piece of cake.
45 degrees is exactly half of 90 degrees. Since the two non-right angles in a right triangle add to 90 degrees, that means that if one angle is 45 degrees so is the other- a 45, 45, 90 degree triangle is isosceles. If one leg has length, say, 1, so does the other. By the Pythagorean theorem, the length of the hypotenuse, c , is given by $\displaystyle c^2= 1^2+ 1^2= 2$ so $\displaystyle c= \sqrt{2}$. Now, sine is "opposite side/hypotenuse" so $\displaystyle sin(45)= \frac{1}{\sqrt{2}}$ and, "rationalizing the denominator, that is $\displaystyle \sin(45)= \frac{1}{\sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}}= \frac{\sqrt{2}}{2}$.