Transcript 5 2 Verify Trig Identities

Objective:
1. To verify trig
identities
Assignment:
Error Analysis:
• P. A70-A71: Read
• P. A75: 3, 7, 9, 11, 12,
17
• P. 387: 1-38 S
• P. 388: 57, 58
An identity in mathematics is a type of equation
that is true for all values within the domain of
the variable.
For example, sin2 x + cos2 x = 1 is an identity
since it is true no matter what real number
you input for x.
Determine whether or not the equation below is
an identity.
1
cos x 
2
Even though this equation has an
infinite number of solutions, it is
not an identity since it is not true
for every possible value of x.
This is called a conditional
equation, and it’s the type you
are most used to solving.
Conditional equations and identities are both
types of equations, but you solve them
differently. As a matter of fact, you don’t
solve an identity at all.
When you solve a conditional equation, you use
properties of algebra and real numbers to
move quantities from one side to the other to
find the value or values that make the
equation true.
An identity, however, is already true for all
values (within the domain). Your job is to
verify that it is, in fact, true.
You don’t move stuff from one side of the
equation to the other. Your goal is to make
one side look like the other side through
substitution and simplification.
Think of your job like this:
You have a trig expression that you need to
simplify. As your book frustratingly points out,
there’s often “more than one form of each
answer.” Verifying an identity can lessen that
frustration since it provides the answer you
are looking for. You just have to find all the
right steps to get there.
These “guidelines” are straight from your book, and they
actually offer some sound advice.
Verify the identity:
sin 2   cos 2 
1
2
2
cos  sec 
Verify the identity:
sec2   1
2

sin

2
sec 
Verify the identity:
1
1

 2 csc2 
1  cos  1  cos 
Verify the identity:
1
1

 2sec 2 
1  sin  1  sin 
Verify the identity:



sec 2 x  1 sin 2 x  1   sin 2 x
Verify the identity:



tan 2 x  1 cos 2 x  1   tan 2 x
Verify the identity:
csc x  sin x  cos x cot x
Verify the identity:
tan x  cot x  sec x csc x
How could you simplify the expression below so
that there was only a monomial in the
denominator?
sin 
1  cos 
If you want to “turn” a
binomial “into” a
monomial when it is in the
denominator of a fraction,
just multiply by the
“conjugate.”
Verify the identity:
sin 
csc   cot  
1  cos 
Verify the identity:
cos y
sec y  tan y 
1  sin y
When both sides of an identity are overly
complicated, then you can try to work on both
sides simultaneously.
Don’t move anything from one side of the
equation to the other!
You’re just trying to show that both sides
simplify down to the same expression.
Verify the identity:
tan 2 
1  cos 

1  sec 
cos 
Verify the identity:
cot 2 
1  sin 

1  csc 
sin 
Objective:
1. To verify trig
identities
Assignment:
Error Analysis:
• P. A70-A71: Read
• P. A75: 3, 7, 9, 11, 12,
17
• P. 387: 1-38 S
• P. 388: 57, 58