Relationships of negative angles

You can convince yourself of these through the use of the unit circle and
talking about an initial reference angle theta that is negative. For example
show that sin(-2π/3) = -sin(2π/3)
sin( )   sin 
csc( )   csc 
tan( )   tan 
cos( )  cos 
sec( )  sec 
cot( )   cot 
The equation of a circle is x2+y2=r2
 Because the unit circle has r=1 and each x value is the same
as cos(Ө) and each y value is the same as sin(Ө) we get the
following idea.

sin   cos   1
2

2
This is true not only because of the equation of a circle but
because of the Pythagorean Theorem
This is called a PYTHAGOREAN IDENTITY
2 other relationships can occur through simple division.
sin   cos   1
2
2

Start with the above statement and divide both
sides by sin2Ө

Now start with the same thing and divide both
sides by cos2Ө

These are also called Pythagorean Identities
Cofunctions
Sin
Cos
 Sec
Csc
 Tan
Cot
 Notice the names of these things can pair
themselves up into groups of 2, we refer to
them as cofunctions, what follows is the
explanation of what cofunctions are and how
they are related to one another.

Lets say that angle A=35 degrees.
A
Then the following would be true.
o
35
Sin(35)=a/c
b
c
Tan(35)=a/b
o
55
C
a
What would Cos(B), and Cot(B) end up being?
B Well we should know that B=90-35 or 55
degrees.
Therefore;
Cos(55)=a/c
Cot(55)=a/b
Notice that the 2 cofunctions shown (sine/cosine and tangent/cotangent) though
they are of different degrees the ratios for sine and cosine are the same and the
ratios for tangent and cotangent are the same.
Does it make sense that the sin(A)=cos(B)?

Basically what we are saying is that angle A and
angle B are complementary angles so whatever
one angle is, the other is 90-(the first angle).

We will refer to A as the angle we start with and
B=90-A

This provides the better notations below…
sin A  cos(90o  A)
tan A  cot(90o  A)
sec A  csc(90o  A)
Do not need to write in your notes
Identities


You know some basic algebraic identities
(equivalent statements that allow you to re-write
one expression as another without making
numerical changes, we only change the
appearance and not the numerical value). For
example you know that (a+b)2=a2+2ab+b2. That
is an identity. The difference of perfect squares
is another algebraic identity that you are familiar
with.
In this course we are going to work with
trigonometric identities. Our goal is to take a
complicated expression and use substitution and
creativity to re-write the expression as a simpler
form without changing numerical values.
Here are some identities that you should be familiar with so far…
The cofunction Identities for whatever reason do not show up very often.
However we will be adding to this list.
Do not need to write in your notes
There are two styles of problems that you
most generally come across. One is to
simplify an expression. The other takes
two expressions one being more complex
than the other, sets them equal to one
another and asks you to prove that the
more complex expression is actually equal
to the simpler expression. In all actuality
the process for each of the 2 styles is
pretty much the same.
 The purpose of doing this is so that we
acquire an ability to solve trigonometric
equations which is often an integral part of
any calculus course.

A helpful hint:

If you get stuck try putting everything into terms of
sin and cos.
Look hard for Pythagorean identities or forms of
the Pythagorean identities. For example I would
2
2
like to find sin   cos  in my expressions
because it can be replaced with 1.
 Other forms that are useful

sin   1  cos 
2
2
cos   1  sin 
2
2
Simplify sec x  sin x tan x

We do not recognize any Pythagorean Identities. So
transfer everything over into terms of sin and cos.
1
sin x
 sin x 
cos x
cos x

Hints: Cancel things that cancel…Re-write expressions with
common denominators. USE ALGEBRA (nothing cancels)
2
1
sin x

cos x cos x
1  sin x
cos x
2
2
cos x
cos x
cos x
cos t  tan t sin t
sin x
cos x

cos x 1  sin x
Do you recognize any quotient
identities?
 Does that help any?
 Try re-writing as one fraction.

sin x sec x
tan x
cos x  sin x cos x
3
2
The following questions ask you to prove or verify an
identity. Do the same thing as before, but may have
to work on both sides of the equation.