Download Section 7.1 Basic Trig. Identities

WHAT IS AN IDENTITY?
A rule that can be used…. Most of the time forward as well as backwards
Two functions f and g are said to be identically equal if
Pre-Calculus
Chapter 7
f (x) = g(x)
Section 1
Basic Trigonometric Identities
for all values of x
for every value of x for which both functions are defined.
Such an equation is referred to as an identity. An equation
that is not an identity is called a conditional equation.
Example ( non trig that is)
x  y  ( x  y)( x  y)
2
2
They are only identities if we can prove or explain them.
Prove that sin(x)tan(x) = cos(x)
is not a trigonometric identity by producing
a counterexample.
Suppose x = /6
 6 tan 6  ?  ? cos 6 
sin 
1 2 tan
3
 ?  ? cos 3 
 2
3 


not equal
Go ahead try it….
Another one if needed:
Prove that sin(x)cos(x) = tan(x)
is not a trigonometric identity by producing a counterexample.
Suppose x = /4
Notes:
1.) Producing a counterexample can show that an equation is
not an identity. Proving that an equation is and identity
generally takes more work. It is the showing that ALL values
of the defined variable will be the challenge.
2.) I need to encourage you to start creating a careful and well
organized list of all of the trigonometric identities. We will
continue to accumulate this list as we continue with the
trigonometry in the next trimester.
At times I will allow you to use this list on quizzes and test….. Not all 
rules… if must be in your own handwriting and on lined paper NO
example problems just the identities as given in our book or as agreed
upon as a class (both classes)
Suggestions:
1. Pick common angles you know from the unit circle.
2. Your basic number knowledge will help
(adding, dividing, positives/ negatives, etc.)
Examples 1:
Prove that each equation is not a trigonometric identity by
producing a counterexample
A.)
sec 
 sin 
tan 
Let us try
0,
B.)
sec 2   1 
   
, , , , etc.
6 4 3 2
until we find one
Yes there will be other methods…. PROOF’s
Sample answers a.)

4
b.)
cos 
csc 

Stop and do some examples
6
1
Examples 2:
use the given information to find the trigonometric value.
3
4
cos  
A.) if
Since
then find
sec  .
Examples 2:
use the given information to find the trigonometric value.
B.) if
1
sec  
cos 
tan  
tan 2   1  sec 2 
since
tan  
then find
.
cos 
4
 cos 
19
2
 19
 sec 
4
Alternative method
3
4
cos 
then find
 3
2


 4   1  sec 


19
 sec 2 
16
use the given information to find the trigonometric value.
B.) if
3
4
Use the Pythagorean Identity
1
By substitution sec  
3
 
4
4
So sec  
3
Examples 2:
tan  
.
opp
y

adjacent x
We can draw in the reference triangleS with those
side lengths. You need to pay attention to signs and
thus quadrants this reference triangle may be in.
Probably 2 locations. {alright it is more of a circle
than the unit circle if you really get it}
Case 2 is a rotation of 180
Case 3 x axis reflection
Case 4 y axis reflection
Leave up on screen as you do examples on the next slide
or the unit circle can address this……
Alternative method
Examples 2:
use the given information to find the trigonometric value.
B.) if
Since
tan  
tan  
3
4
, then find
cos 
.
Diagram
opposite y  y
 
adjacent x  x
Not to scale
r
y
-x
And
x2  y2  r 2
(4) 2  ( 3 ) 2  r 2
19  r 2
19  r
cos  
Case 1 is really a rotation of 360
 4 19
 cos 
19
 4  4 19

19
19
-y
x
r
Thus use the Pythagorean Theorem to
find the missing side.
Wait isn’t that what we said in the
previous method???
Pythagorean Identifdy…???
some examples like page 425 example 3
Express each value as trig function on an angle in the 1 st Quadrant
cos(750)
 35 
sin 

 6 
cos(930)
 25 
tan 

 4 
( one more identity slide to go)
2
Explanation #1
some examples like page 425 example 3
Express each value as trig function on an angle in the 1 st Quadrant
 35 
sin 

 6 
Express each value as trig function on an angle in the 1 st Quadrant
cos(750)  cos 30
 35    sin  
sin 

6
 6 
  sin 30
cos(930)
 25   tan   
tan 
 

4
 4 
Some times I find it easier to draw a circle centered on the origin.
Draw in the angle given. Note the quadrant you end in and the
reference angle.
See next slide for why/work on
 35 
sin 

 6 
 24 11 
sin 


6 
 6
 11 
sin 

 6 
 
  sin  
6
11
6
 
  sin  
6
Since they are conterminal
but you are in the wrong quadrants
we need to move it into the 1st quadrant in our rewrite
Flipping it over the x axis allows the sine’s to be opposites
( one more identity slide to go)
Doing algebra with trig functions
Express each value as trig function on an angle in the 1 st Quadrant
 35 
sin 

 6 
The answer
 
  sin  
6
Case 3 of the symmetric identities
 
  sin  
6
6
( one more identity slide to go)
Explanation #2 using case 4 of the symmetric identities
 36  
sin 
 
6
 6
The answer

sin(2k  A)   sin A
Simplify each expression….
1.)
2.)
cot( )
cos( )
 cos( ) 
 sin( ) 


cos( )
cos( )
1

sin( ) cos( )
csc( )
sin(   )
cos(   )
Quotient identities
 sin( )
 cos( )
Symmetric
identity
case 2
dividing by a fraction
tan( )
quotient identities
Reducing your fractions
( one more identity slide to go)
More Examples like those on page 426 example 4
Simplify each expression….
3.)
csc x  cos x cot x
1
cos x
 cos x
sin x
sin x
1
cos 2 x

sin x
sin x
4.)
sin x  sin x cot 2 x
sin x(1  cot 2 x)
sin x(csc2 x)
sin 2 x
sin x
 1 
sin x 2 
 sin x 
1
sin x
sin x
csc x
1  cos 2 x
sin x
3