Download 1-6

1997 BC Exam
1.6 Trig Functions
Photo by Vickie Kelly, 2008
Black Canyon of the Gunnison
National Park, Colorado
Greg Kelly, Hanford High School, Richland, Washington
Answer as quickly as you can!
First, a little review.
y


2
 3
tan 
 4

  1

cot    undefined



3
2

0
2
x
 3
csc 
 4



2
 
tan    1
4
 5
sec 
 4

  2

 3
cos 
 4
1




2


Answer as quickly as you can!
First, a little review.
y


2
 3
sin 
 4



3
2

0
2
x



1
2
 3 
sec 
  2
 4 
1
 7 
cos 


2
 4 
 3
sin 
 2

  1

 5
csc 
 4

  2

sin  0  
0

Trigonometric functions are used extensively in calculus.
When you use trig functions in calculus, you must use radian
measure for the angles.
To check or change the angle mode:
Press:
5
2
Settings
Document Settings
Make sure you set the angle
mode to Radian, then scroll
down and click Make Default.
You could also click Restore,
which returns the calculator to
the factory settings, which
include radian mode, and then
click Make Default.

The best plan is to leave the calculator mode to radians and
o
use 
when you need to use degrees.
To find trig functions on the TI-nspire, press
desired function, and press enter .
trig
, select the
If you want to brush up on trig functions, they are graphed in
your book.

Even and Odd Trig Functions:
“Even” functions behave like polynomials with even
exponents, in that when you change the sign of x, the y
value doesn’t change.
Cosine is an even function because: cos     cos  
Secant is also an even function, because it is the reciprocal
of cosine.
Even functions are symmetric about the y - axis.

Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd
exponents, in that when you change the sign of x, the
sign of the y value also changes.
Sine is an odd function because:
sin      sin  
Cosecant, tangent and cotangent are also odd, because
their formulas contain the sine function.
Odd functions have origin symmetry.

The rules for shifting, stretching, shrinking, and reflecting the
graph of a function apply to trigonometric functions.
Vertical stretch or shrink;
reflection about x-axis
a  1 is a stretch.
Vertical shift
Positive d moves up.
y  a f b  x  c   d
Horizontal shift
Horizontal stretch or shrink;
Positive c moves left.
reflection about y-axis
b  1 is a shrink. The horizontal changes happen
in the opposite direction to what
you might expect.

When we apply these rules to sine and cosine, we use some
different terms.
A is the amplitude.
Vertical shift
 2

f  x   A sin   x  C    D
B

Horizontal shift
B is the period.
B
4
A
3
C
2
D
1
-1
0
-1
 2

y  1.5sin   x  1   2
 4

1
2
x
3
4
5

Trig functions are not one-to-one.
However, the domain can be restricted for trig functions
to make them one-to-one.

2 
y  sin x
3
2


2

2
3
2

2
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and
ranges are defined in the book.

You will be using trig identities throughout the year to solve
calculus problems.
Today we will look at some of those identities and where
they come from.
When you need to use a trig identity you will not have time
to generate the identity from scratch. They need to be
memorized!

The easiest trig identity is the Pythagorean Identity:
y
 cos ,sin  
A

O
Since the hypotenuse of
this triangle has a
x length of one, we can
1,0  just use the
Pythagorean Theorem:
sin 2   cos 2   1

Consider angles u and v in standard position on the unit
circle, determining points A and B and their coordinates:
y
 cos v,sin v 
We could find the length of
chord AB by using the
distance formula:
v
B
u

x
O
A
 cos u,sin u 
AB 
 cos v  cos u    sin v  sin u 
Let the difference between the angles be:
2
  u v
2

We could rotate angle AOB around to standard position
without changing the length of chord AB:
y
 cos v,sin v 
B

x
O
A
 cos u,sin u 
AB 
 cos v  cos u    sin v  sin u 
2
2

We could rotate angle AOB around to standard position
without changing the length of chord AB:
y
 cos ,sin  
A
We rewrite the coordinates
of A and B in terms of  :

x
O
B 1,0
 
Using the distance formula:
AB 
 cos  1   sin   0 
2
Since the lengths of the chords are the same, we can
set the two expressions equal to each other.
2

 cos v  cos u    sin v  sin u 
2
2

 cos v  cos u    sin v  sin u 
2
2
 cos   1   sin   0 
2
  cos   1   sin  
2
2
2
cos 2 v  2cos u cos v  cos 2 u  sin 2 v  2sin u sin v  sin 2 u
 cos 2   2cos   1  sin 2 
1  2 cos u cos v  1  2sin u sin v  1  2 cos   1
2  2 cos u cos v  2sin u sin v  2  2 cos 
1  cos u cos v  sin u sin v  1  cos 
 cos u cos v  sin u sin v   cos 
cos   cos u cos v  sin u sin v
  u v
cos  u  v   cos u cos v  sin u sin v

cos  u  v   cos u cos v  sin u sin v
Starting from this formula we can find a similar identity:
cos  u  v   cos  u   v  
cos  u  v   cos u cos  v   sin u sin  v 
Cosine is an even function, and sine is an odd function:
cos  u  v   cos u cos v  sin u sin v
For convenience, we combine the two formulas like this:
cos  u  v   cos u  cos v sin u  sin v
These symbols must be written correctly!

The co-function identities are simple to find from the triangle:
For example:
adjacent
y


cos     

2
 hypotenuse z
opposite
y
sin  

hypotenuse z

z
2


y


sin   cos    
2

x
The co-function identities are not actually included on the
calculus quizzes, but they are useful.



sin   cos    
2



sin  u  v   cos    u  v  
2

 
 
sin  u  v   cos    u   v 
 
 2
cos  u  v   cos u cos v  sin u sin v




sin  u  v   cos   u  cos v  sin   u  sin v
2

2

sin  u  v   sin u  cos v  cos u  sin v
Using the properties of
odd and even functions:
sin  u  v   sin u  cos v  cos u  sin v

There are sixteen trig identities
on the calculus formula sheets.
Starting with the formulas in this lecture, you should
be able to derive the others for practice, or for fun!
These formulas are sometimes difficult to
remember, so if you haven’t already you should
make flashcards and get started memorizing!
