Special Right Triangle
45- 45- 90
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The 45- 45- 90 triangle is one of two special right triangles we will be investigating. 
The "special" nature of these triangles is their ability to yield exact answers instead of decimal approximations when dealing with trigonometric functions.

If we represent the legs of an
 isosceles right triangle by 1, we can use the Pythagorean Theorem to establish pattern relationships between the lengths of the legs and the hypotenuse.  These relationships will be stated as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means.

There are two pattern formulas that apply ONLY to the 45-45-90 triangle.

Note:  the legs need not be a length of 1 for these patterns to apply.
The patterns will apply with any length legs.
 

45-45-90 (Isosceles Right Triangle)
Pattern Formulas
(you do not need to memorize these formulas as such, but you do need to memorize the patterns)



H = hypotenuse
L = leg

 

 

What if I forget the pattern formulas?
What should I do?

The nice thing about mathematics is that there is always another way to do the problem.  If you forget these formulas, you could always use the Pythagorean Theorem or a trigonometric formula.

Let's look at 3 solutions to this problem where you are asked to find x:

Pattern Formula solution Pythagorean Theorem solution Trigonometric solution

We are looking for the hypotenuse so we will use the pattern formula that will give the answer for the hypotenuse:

Substituting the leg = 7, we arrive at the answer:

A nice feature of the pattern formulas is that the answer is already in reduced form.

Since a 45-45-90, also called an isosceles right triangle, has two legs equal, we know that the other leg also has a length of 7 units.
c2 = a2 + b2
x2 = 72 +72
x2 = 49 + 49
x2 = 98


Use either 45 angle as the reference angle.  One possible solution is shown below:

rounded
rounded


Using the newly found patterns in trig problems:

1.  Find the exact value of
           sin 45 + cot 45.

Solution:
          

 

 

2.  Find the exact value of
           (csc 45)(cos 45).

Solution: