Download The Trigonometric Functions

The
Trigonometric
Functions
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 old Indian 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
csc  
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
Computing the Values of Trig
Functions of Acute Angles
TRIANGLES
The 45-45-90 Triangle
In a 45-45-90 triangle the sides are in a ratio of 1- 1- 2
This means I can build a triangle with these lengths for sides
(or any multiple of these lengths)
We can then find the six trig
functions of 45° using this triangle.
o
1
2

sin 45  
rationalized
h
2
2
a
1
2
cos 45  

h
2
2
0 1
tan 45    1
a 1
45°
2
1
45°
Can "flip" these to
get other 3 trig
functions
90°
1
You are expected to know exact values for trig functions of 45°.
You can get them by drawing the triangle and using sides.
What is the radian equivalent of 45°?

4
You also know all the trig
functions for /4 then.
45°

2
sec 
 2
1
4
reciprocal of cos so h over a

1
tan
 1
4 1
2
1
45°
90°
1
The 30-60-90 Triangle side opp 60°
In a 30-60-90 triangle the sides are in a ratio of 1- 3 - 2
side opp 90°
side opp 30°
This means I can build a triangle with these lengths for sides
We can then find the six trig functions of 30°or 60°
using this triangle.
I used the triangle and
o 1
sin 30  
h 2
a 1
cos 60  
h 2
did adjacent over
hypotenuse of the 60°
to get this but it is the
cofunction of sine so
this shows again that
cofunctions of
complementary angles
are equal.
30°
2
3
60°
90°
1
o 1
3 Be sure to locate the angle you want
tan 30  

a
3 before you find opposite or adjacent
3
What this means is that if you memorize the special triangles,
then you can find all of the trig functions of 45°, 30°, and 60°
which are common ones you need to know.
You also can find the radian equivalents of these angles.
45 

4
30 

6
60 

3
You need to know all of the values in Table 3 on
page 520 in your book and special triangles can
help you with that.
When directions say "Find the exact value", you must
know these values not a decimal approximation that
your calculator gives you.
Using a Calculator to Find
Values of Trig Functions
If we wanted sin 38° we could not use the
previous methods to find it because we
don't know the lengths of sides of a
triangle with a 38° angle. We will then
use our calculator to approximate the
value.
You can simply use the sin button on
the calculator followed by (38) to find
the sin 38°
A word to the wise: Always make sure your calculator
is in the right mode for the type of angle you have
(degrees or radians)