What are the basic properties of a 45-45-90 triangle?
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The basic properties are: An interior angle of ##45^@## an exterior angle of ##135^@## an opposite angleof ##45^@## a right angle (##90^@##) opposite side length: ##sqrt2/2## and adjacent side length of ##sqrt2/2## and a hypotenuse of 1. (side lengths for a standard triangle inside of the Unit Circle)
Explanation
All standard right triangles inside of the unit circle have a hypotenouse of 1. This is because the unit circle is defined as a circle with a radius of 1. Thus we create Unit Circle Triangles by picking a radius and drawing its vector components.
In this case we know we have a triangle with a primary angle of ##45^@## and as mentioned before a hypotneouse of 1. As such we can use ##sintheta## and ##costheta## to determine the other two lengths.
##sintheta=o/h->hsintheta=o->1*sin(45^@)=sqrt2/2##
##costheta=a/h->hcostheta=a->1*cos(45^@)=sqrt2/2##
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