what is the significance between cos(45) being equal to sqrt(0.5)

Question

I was messing with a calculator and found that the square root of 0.5 is equal to the sine and cosine of 45. I was just wondering the correlation between the two equations.

Answer 1

if you consider a right triangle in the first quadrant, with hypotenuse of length 1, you realize that it's isosceles so each leg measures 1/√2.  Now it's obvious that the cosine (and sine, too) of the acute angles is this value 1/√2.

Answer 2

Draw a 45-45-90 right triangle with sides = 1.  By the Pythagorean Theorem, the hypotenuse is √2.  By definition, we have:

 

sin(45) = cos(45) = 1/√2

 

Now rationalize the denominator:

 

1/√2 = 1/√2 * √2/√2 = (1/2)√2 = √(2/4) = √(1/2)