Transcript Right Triangle Triconometry

Right Triangle Trigonometry
M 140 Precalculus
V. J. Motto
In a right triangle, the shorter sides are called legs and the longest side
(which is the one opposite the right angle) is called the hypotenuse
We’ll label them a, b, and c and the angles 
and . Trigonometric functions are defined by
taking the ratios of sides of a right triangle.

adjacent
c
First let’s look at the three basic functions.
leg
b
SINE

leg a
COSINE
TANGENT
They are abbreviated using their first 3 letters
opposite
a
opposite a
sin  

tan  

hypotenuse c
adjacent b
adjacent
b
cos  

hypotenuse c
We could ask for the trig functions of the angle  by using the definitions.
You MUST get them memorized. Here is a
mnemonic to help you.

c
The sacred Jedi word:
b
SOHCAHTOA

adjacent
a
opposite
b
sin  

hypotenuse c
adjacent
a
cos  

hypotenuse c
opposite b
tan  

adjacent a
It is important to note WHICH angle you are talking
about when you find the value of the trig function.

c
5
4b
Let's try finding some trig functions
with some numbers. Remember that
sides of a right triangle follow the
Pythagorean Theorem so
a b c
2

2
2
adjacent
a3
sin  =
o 3

h 5
Let's choose:
tan  =
o 4

a 3
32  4 2  5 2
Use a mnemonic and
figure out which sides
of the triangle you
need for tangent.
sine.
You need to pay attention to which angle you want the trig function
of so you know which side is opposite that angle and which side is
adjacent to it. The hypotenuse will always be the longest side and
will always be opposite the right angle.

Oh,
I'm
acute!
This method only applies if you have
a right triangle and is only for the
acute angles (angles less than 90°)
in the triangle.
5
4

3
So
am I!
There are three more trig functions. They are called the
reciprocal functions because they are reciprocals of the first
three functions.
Oh yeah, this
Like the first three trig functions, these are referred
to by the first three letters except for cosecant since
it's first three letters are the same as for cosine.
opposite
sin 
hypotenuse
means to flip the
fraction over.
hypotenuse
cosecant 
opposite
adjacent
cos 
hypotenuse
hypotenuse
secant 
adjacent
opposite
tan 
adjacent
adjacent
cotangent 
opposite
Best way to remember these is learn which is reciprocal of which and flip them.
5
sec  =
4
Let's try one:

a
h
c
5
Which trig function is this the reciprocal of?
adjacent
hypotenuse
so sec is
cos is
hypotenuse
adjacent
4b
o
cot  =
a
a3

3
4
opposite
adjacent
tan is
so cot is
adjacent
opposite
As a way to help keep them straight I think, The "s" doesn't go with "s" and
the "c" doesn't go with "c" so if we want secant, it won't be the one that
starts with an "s" so it must be the reciprocal of cosine. (have to just
remember that tangent & cotangent go together but this will help you with
sine and cosine).
TRIGONMETRIC IDENTITIES
Trig identities are equations that are true for all angles in the domain. We'll be
learning lots of them and use them to help us solve trig equations.
RECIPROCAL IDENTITIES
These are based on what we just learned.
1
cosec 
sin 
1
sec  
cos 
1
cot  
tan 
We can discover the quotient identities if we take quotients of sin and cos:
0
o
sin  h o  h
  tan 
 
cos  a h a
a
h
Which trig function is this?
Remember to simplify
complex fractions you invert
and multiply (take the bottom
fraction and "flip" it over and
multiply to the top fraction).
Try this same thing with
what do you get?
cos 
sin 
and
QUOTIENT IDENTITIES
These are based on what we just learned.
sin 
tan  
cos 
cos 
cot  
sin 
Now to discover my favorite trig identity, let's start with a right triangle and
the Pythagorean Theorem.
a 2  b2  c2
b a c
c2 c2 c2
2
2
b a
     1
c c
2
c
h
b
o
a

a
This one
is sin
2
2
This one
is cos
Rewrite trading terms places
Divide all terms by c2
Move the exponents to the outside
sin  2  cos  2  1
Look at the triangle and the angle  and determine which trig function these are.
sin   cos   1
2
2
This is a short-hand way you can write trig functions that are squared
Now to find the two more identities from this famous and often used one.
sin   cos   1
2
2
cos2
cos2
What trig function
is this squared?
1
Divide all terms by cos2
cos2
What trig function
is this squared?
tan   1  sec 
2
2
sin   cos   1
2
sin2
1
2
sin2
Divide all terms by sin2
sin2
What trig function What trig function
is this squared?
is this squared?
1  cot   cosec 
2
2
These three are sometimes
called the Pythagorean
Identities since they come
from the Pythagorean
Theorem
RECIPROCAL IDENTITIES
1
cosec 
sin 
1
sec  
cos 
1
cot  
tan 
QUOTIENT IDENTITIES
cos 
sin 
cot  
tan  
sin 
cos 
PYTHAGOREAN IDENTITIES
sin   cos   1
2
2
tan   1  sec 
2
2
1  cot 2   cosec 2
All of the identities we learned are found in the back page of your book under
the heading Trigonometric Identities and then Fundamental Identities.
You'll need to have these memorized or be able to derive them for this course.
If the angle  is acute (less than 90°) and you have
the value of one of the six trigonometry functions,
you can find the other five.
Reciprocal of sine so "flip" sine over
Sine is the ratio of which
sides of a right triangle?
o
1
sin   
3
h
cosec  3
a 1  3
2

1
2
When you know 2 sides of a right
triangle you can always find the 3rd
with the Pythagorean theorem.
a 82 2
3
2 a2
2
a 2 2
cos  
3
h
Draw a right triangle and
label  and the sides you
know.
o
1
tan   
a 2 2
Now find the other
trig functions
sec  3
2 2
cot  2
"flipped"
cos
"flipped"
2 tan
There is another method for finding the other 5 trig
functions of an acute angle when you know one function.
This method is to use fundamental identities.
We'd still get cosec by taking reciprocal of sin
1
sin  
3
cosec  3
sin   cos   1
2
2
Now use my favourite trig identity
Sub in the value of sine that you know
2
1
2
Solve this for cos 
   cos   1
3
This matches the
8
2
2
8
2
answer we got with
cos  

cos  
the other method
3
9
9
square root
We won't worry about 
both sides
because angle not negative
You can easily find sec by taking reciprocal of cos.
Let's list what we have so far:
1
sin  
3
3
2 2
sec  
cosec  3 cos  
3
2 2
sin 
We need to get tangent using
tan  
fundamental identities.
cos 
1
3
Simplify by inverting and multiplying
1
3
1
tan  
 

2 2
3 2 2 2 2
3
cot   2 2
Finally you can find
cot by taking the
reciprocal of this
answer.
SUMMARY OF METHODS FOR FINDING THE
REMAINING 5 TRIG FUNCTIONS OF AN ACUTE
ANGLE, GIVEN ONE TRIG FUNCTION.
METHOD 1
1. Draw a right triangle labeling  and the two sides you
know from the given trig function.
2. Find the length of the side you don't know by using
the Pythagorean Theorem.
3. Use the definitions (remembered with a mnemonic) to
find other basic trig functions.
4. Find reciprocal functions by "flipping" basic trig
functions.
METHOD 2
Use fundamental trig identities to relate what you know
with what you want to find subbing in values you know.
The sum of all of the angles in a triangle always is 180°
What is the sum of  + ? 90°
Since we have a 90° angle, the sum of the other two angles
must also be 90° (since the sum of all three is 180°).

Two angles whose sum is
90° are called
complementary angles.
c
b
a
a
What is sin  ? c
a
What is cos ?
c

Since  and  are
complementary angles and
sin  = cos ,
sine and cosine are called
cofunctions.
This is where we get the name
cosine, a cofunction of sine.
Looking at the names of the other trig functions can
you guess which ones are cofunctions of each other?
secant and cosecant
tangent and cotangent
Let's see if this is right. Does sec  = cosec ?
hypotenuse over adjacent

c
b
a

hypotenuse over
opposite
c
sec    cosec
b
This whole idea of the
relationship between
cofunctions can be
stated as:
Cofunctions of complementary
angles are equal.
Cofunctions of complementary angles are equal.
cos 27° = sin(90° - 27°) = sin 63°
Using the theorem above, what trig function of
what angle does this equal?
  cot       cot 3 


 
tan
8
2 8
 8 
Let's try one in radians. What trig functions of
what angle does this equal?

The sum of complementary angles in radians is
2
90° is the same as 
2

Basically any trig function then equals 90° minus or 2
minus its cofunction.
since
sin 36 sin 36

 tan 36
sin 54 cos 36
We can't use fundamental identities if the trig functions are
of different angles.
Use the cofunction theorem to change the denominator
to its cofunction
Now that the angles are the same we can use a trig
identity to simplify.