Special Right Triangle
30- 60- 90
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The
30- 60- 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 you draw an altitude in an equilateral triangle, you will form two congruent 30- 60- 90 triangles.  Starting with the sides of the equilateral triangle to be 2, the Pythagorean Theorem will allow us to establish pattern relationships between the sides of a 30- 60- 90 triangle.  These relationships will be stated here 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 three pattern relationships that we can establish that apply ONLY to a 30-60-90 triangle.

Note:  the hypotenuse need not be a length of 2 for these patterns to apply.
The patterns will apply with any length hypotenuse.
 

30-60-90 Triangle Pattern Formulas
(you do not need to memorize these formulas as such, but you do need to memorize the relationships)

        

Labeling:

H = hypotenuse
LL
= long leg (across from 60)
SL
= short leg (across from 30)

 


Short Cut Pattern Formulas:
(These formulas give answers directly.
We don't need to work out the trig,
as we already know the pattern relationships)

short leg:

You must remember that these formula patterns can be used ONLY in a 30-60-90 triangle.

long leg:

combining the first two:

 

 

 

Using the patterns to find the lengths of sides:

EASY:

Find x and y.

x is the short leg



  Answer

y is the long leg



 Answer

 

 

HARDER:

Find x and y.

6 is the short leg and
x is the hypotenuse
(start with what you have given)



  Answer

y is the long leg



 Answer

 

 

HARDER:
(requires more algebraic manipulation)

Find x and y.

8 is the long leg and 
x is the hypotenuse
(start with what you have given)





  Answer

y is the short leg




 Answer



Using the newly found patterns in trig problems:

1.  Find the exact value of
           tan 30 + cos 60.

Solution:
          

 

 

2.  Find the exact value of
                 (sec 30)2.

Solution:
   

 

 

What if I forget the formula patterns?
What should I do?

There is always more than one way to tackle a problem.  If you forget these formula patterns, you could always use the trigonometry formulas to find one of the missing sides and then use the Pythagorean Theorem to find the last side OR you could use the trigonometry formulas to find both missing sides.

Unfortunately, the Pythagorean Theorem by itself,
will not help you find both of the missing sides.  Remember that you need to know TWO sides of a triangle in order to engage the Pythagorean Theorem.