How do you calculate the sine, cosine and tangent values for 45 degrees without a calculator?

To find the values for 45 degrees, we construct a right angled isosceles triangle, which has two sides of equal length. We can say that these sides are 1 unit long, and so we can figure out the value of the hypotenuse of this triangle using Pythagoras' theorem. Once we have figured out that the hypotenuse is sqrt(2), we can then figure out the sine, cosine and tangent values using what we know about SOHCAHTOA. To find sine, we divide the opposite side to the 45 degreee angle, which has a length of 1 unit by the hypotenuse which has a value of sqrt(2) to find that sin(45) is equal to 1/sqrt(2) (which we are able to rationalise to sqrt(2)/2 ). Since this triangle is an isosceles, we know that the opposite and adjacent sides are of equal length, and so we can see the cos(45) is also equal to 1/sqrt(2). In order to find out the tangent value of 45 degrees, we need to divide the opposite side to the angle by the adjacent side. Since both the opposite and adjacent sides of this triangle are the same, we can see that tan(45) is equal to 1.