Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.3
Page 1 / 3
Even-Odd Properties of Trig functions
Draw a unit circle, with P = (a,b) on it, which is where we get to using an angle of θ.
Then draw the arc for –θ, and P' = (a, -b).
(In other words, it's just flipped around the X axis)
Going back to the various definitions, we see that (for example)
If Sinθ = b, then Sin-θ = -b  Sin –θ = - ( Sin θ ), which gets us our first mini-proof, below
Sin    Sin 
Cos    Cos 
Tan    Tan 
Csc    Csc 
Sec    Sec 
Cot    Cot 
§7.3 – Establishing Trig Identities
What is an identity?
A mathematical identity says that two functions (named f and g) are identically equal if, for
any x we could possibly choose, f(x) = g(x)
 In other words, NO MATTER WHAT VALUE WE CHOOSE FOR X, f(x) = g(x)
This is different from conditional equations, such as 2x = 6.
 This is only true when x is 3
 For conditional equations in math classes, they often lead directly into "solve for x" sorts of
problems.
How can we establish new identities?
We want to show that some expression is identically equal to a new expression.
 Normally, we'll be given two expressions, and be asked to show that they're identically equal
For example, show that csc  tan   sec
 We will start with one side, and (by a sequence of algebraic manipulations) transform it into
the other side
 Do not treat like solving an equation (do add/subtract/multiply/divide both sides by something)
Example of "Presentation Format":
Given : Establish that csc  tan   sec
csc  tan 
1
sin 
| tanθ =
cos 
sin 
1
sin 

=
sin  cos 
sin 
 1)
 Cancel (
sin 
 Cscθ =
Start with one side. Generally, it's best to start
with the more complicated side, since your job
is then to simplify stuff, rather than make stuff
more complicated
Explain
Rewrite
Explain
Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.3
Page 2 / 3
1
1
= 
1 cos 
1
 1
| 1 • anything = anything
1
1
=
cos 
 Reciprocal identity: 1  Sec
Rewrite
= secθ
Rewrite, and we're finished
Explain
Rewrite
Explain
Cos
So the next question is: What are we allowed to use, in the 'Explain' steps?
 Anything that's mathematically valid (true)
 In particular, the basic trigonometric identities that we've seen previously
 Algebra
In other words, we start with an equation, and we're asked to show that the two sides are
(really) the same thing – that they're identically equivalent.
We do this by starting with one side, transforming (with the explain-rewrite process) until it ends
up looking like the other side.
Basic Trig Identities
So, what steps are we allowed to make?
For starters, anything from the basic trig identities:
Reciprocal:
1
 Csc
Sin
` (and
1
 Sec
Cos
(and
1
 Cos )
Sec
1
 Tan
Cot
(and
1
 Cot )
Tan
Tan 
Sin 
Cos
1
 Sin )
Csc
Quotient:
Cot 
Cos
Sin 
Pythagorean:
Cos2θ + Sin2θ = 1
Cot 2  1  Csc 2
1  Tan 2  Sec 2
Algebra & Trig, Sullivan & Sullivan
Fourth Edition
Notes:§7.3
Page 3 / 3
Also, anything from algebra
I think that a challenge of doing this well will be to remember what you're allowed to do, what
things tend to be good things to do.
I'm going to try talking about a Trig Toolbox as a way of talking about the steps you're allowed
to do.
I'm also going to try talking about Strategies as overall guides for stuff you might want to do.
I HIGHLY, HIGHLY RECOMMEND THAT YOU START KEEPING A LIST OF EACH!!
This will be an invaluable reference tool, as the term goes on!
<Have them do some examples>
Examples:
Basic, with FOIL, common denominator:
#24:
sin  cot   tan    sec
First, let's rewrite the r.h.s. in terms of cos, sin, so we know what we're aiming for: Sec 
Now, let's work with the more complicated left hand side:
= sin   cot   sin   tan 
rewrite in terms of (co)sine
sin   cos  sin   sin 

=
sin 
cos 
simplify:
sin   sin 
= cos  
cos 
Common denominator:
cos   cos  sin   sin 

=
cos 
cos 
Combine, combine exponents:
cos 2   sin 2 
=
cos 
Pythagorean ID:
1
=
cos 
Reciprocal ID:
=sec θ
1
Cos