For the 45°-45°-90° right triangle

Start with a 1x1 square,



It has four right, or 90° angles.

Now cut it into two congruent right triangles by drawing
a diagonal from the lower left to the upper right, like this:



That diagonal bisects the right angles on the lower left
and upper right corners, each into two 45° angles. Now
take away the top triangle, leaving only the bottom half:



Now you have a 45°-45°-90° right triangle.



Use the Pythagorean theorem to calculate its hypotenuse:







So the length of the hypotenuse is 



Now memorize the way this right triangle looks and the 
lengths of the three sides.  For you will need it to find 
the exact values for the sine, cosine, and tangent of 45°:

, (after rationalizing the denominator)

, (after rationalizing the denominator)

[Notice that the sine and cosine of 45° are the same!]

.

--------------------------------------------------------

For the 30°-60°-90° right triangle

Start with an equilateral triangle, each side of
which has length 2,



It has three 60° angles.



Now cut it into two congruent triangles by drawing a median, which
is also an altitude as well as a bisector of the upper 60°-vertex angle:


  
That bisects the upper 60° angle into two 30° angles. It also
bisects the base of the equilateral into two parts, each with
length 1. Now take away the triangle on the right, leaving only 
the one on the left:




Now you have a 30°-60°-90° right triangle.

Use the Pythagorean theorem to calculate its altitude:







So the length of the altitude is 


  
Now memorize the way this right triangle looks and the lengths 
of the three sides.  For you will need it to find the exact
values for the sine, cosine, and tangent of BOTH 60° AND 30°:

,

, 

.

,

, 

,
after rationalizing

[Notice that the sine of 60° and the cosine of 30° are the same,
and also that the sine of 30° and the cosine of 60° are the same!]

Edwin