1997 BC Exam

Photo by Vickie Kelly, 2008
Black Canyon of the Gunnison
National Park, Colorado
Greg Kelly, Hanford High School, Richland, Washington

First, a little review.
 y

2


3
 
2
Answer as quickly as you can!

0
2
 x tan


3
4
  


1 cot
   undefined csc


3

4
 

2 tan

4
1 sec


5

4
 


2 cos


3

4
 


1
2


First, a little review.
 y

2


3
 
2
Answer as quickly as you can!

0
2
 x sin


3

4
 
 sec


3

4
 
 cos


7
4
  
 sin


3

2
 
 csc


5

4
 


1
1

2
2

1
2
2
  
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
Settings
2
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

To find trig functions on the TIn spire, press , 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 a

1 x -axis is a stretch.
Vertical shift
Positive d moves up.
y

    
 d
Horizontal stretch or shrink;
Horizontal shift reflection about y -axis b

1 is a shrink.
Positive c moves left.
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 .
  
A sin


2

B
Vertical shift
  



D
B is the period .
4 B
Horizontal shift
A
3
C
2
D
1 y

1.5sin


2

4
 x

1




2
0 -1 1 2 x
3 4 5

-1

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
   
A
O

1,0 x
 
Since the hypotenuse of this triangle has a length of one, we can 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 ,sin v

B

O u v
We could find the length of chord AB by using the distance formula: x
 cos ,sin u

A
AB
  cos v
 cos u
 sin v
 sin u

2
Let the difference between the angles be:
   v


We could rotate angle AOB around to standard position without changing the length of chord AB: y
 cos ,sin v

B

O x
 cos ,sin u

A
AB
  cos v
 cos u
 sin v
 sin u

2


We could rotate angle AOB around to standard position without changing the length of chord AB: y
   
A
We rewrite the coordinates


O B x
 
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.


 cos v
 cos u
 cos v
 cos u
 sin v
 sin u

2   cos
 
1
  sin
 
0

2
 sin v
 sin u
 cos
 
1
  sin
 
2 cos 2 v
 u v
 cos 2 u
 sin 2 v
 u v
 sin 2 u
 cos 2
 
2 cos
   2

   v
 v
  u v
   
1
 v
 u v
  
 v
 u v
 
 cos cos v
 u v
  cos

 cos
  cos cos v
 u v cos
 u v
 cos cos v
 u v


cos
 u v
 cos cos v
 u v
Starting from this formula we can find a similar identity: cos
  cos
 u
 v
  cos
 u
   

Cosine is an even function, and sine is an odd function: cos
 u
 v
  cos cos v
 u 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: sin
  opposite hypotenuse
 y z z
 
2
 cos
 
2


 adjacent hypotenuse
 y z 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

2
 u v
 sin
 u
 v
  cos

 

2
 u 
 v

 cos
 u v
 sin
 u
 v
  cos

2
 u cos v
 sin

2
 u sin v cos cos v
 u v 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!
