The
30º 60º 90º 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 30º 60º 90º 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 30º 60º 90º 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 30º60º90º triangle. 

Note: the
hypotenuse need not be a length of 2 for these patterns to apply.
The patterns will apply with any length hypotenuse.
30º60º90º
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 60º)
SL = short leg (across from 30º)



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 30º60º90º 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


Using the
newly found patterns in trig problems:
1.
Find the exact value of
tan 30º + cos 60º. 

Solution:


2.
Find the exact value of
(sec 30º)^{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.
