Verifying Trigonometric Identities Math 25 Marie Bruley Merced College.
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Transcript of Verifying Trigonometric Identities Math 25 Marie Bruley Merced College.
Verifying Trigonometric
Identities
Math 25Marie Bruley
Merced College
What is an Identity?
An identity is a statement that two expressions are equal for every value of the variable.Examples:
xxx 2
The left-hand expression always equals the right-hand expression, no matter what x equals.
The fundamental IdentitiesReciprocal Identities Quotient Identities
xx
xSecx
xx
tan
1cot
cos
1sin
1csc
x
xx
x
xx
sin
coscot
cos
sintan
The beauty of the identities is that we can get all functions in terms of sine and
cosine.
The Fundamental IdentitiesIdentities for Negatives
xx
xx
xx
tan)tan(
)cos()cos(
sin)sin(
The Fundamental Identities
xx
xx
xx
22
22
22
csccot1
sec1tan
1cossin
Pythagorean Identities
The only unique Identity here is the top one, the other two can be obtained using the top
identity.
X
Variations of Identities using Arithmetic
Variations of these Identities
xx
xx22
22
sincos1
cossin1
We can create different versions of many of these identities by using arithmetic.
Let’s look at some examples!
Verifying Trigonometric
Identities
Now we continue on our journey!
An Identity is Not a Conditional Equation
Conditional equations are true only for some values of the variable.
You learned to solve conditional equations in Algebra by “balancing steps,” such as adding the same thing to both sides, or taking the square root of both sides.
We are not “solving” identities so we must approach identities differently.
22
4
82
912
2
2
2
xorx
x
x
x
We Verify (or Prove) Identities by doing the following: Work with one side at a time. We want both sides to be exactly the
same. Start with either side Use algebraic manipulations and/or the
basic trigonometric identities until you have the same expression as on the other side.
Example:xxx cossincot
x
xx
x
xx
cos
sinsin
cos
sincot LHS
and xcos RHS
Since both sides are the same, the identity is verified.
Change everything on both sides tosine and cosine.
Suggestions Start with the more complicated side Try substituting basic identities (changing all
functions to be in terms of sine and cosine may make things easier)
Try algebra: factor, multiply, add, simplify, split up fractions
If you’re really stuck make sure to:
Remember to: Work with only one side at a time!
xx
x
xx
cotsin
1cos
csccos RHS
How to get proficient at verifying identities: Once you have solved an identity go back
to it, redo the verification without looking at how you did it before, this will make you more comfortable with the steps you should take.
Redo the examples done in class using the same approach, this will help you build confidence in your instincts!
Don’t Get Discouraged! Every identity is different Keep trying different approaches The more you practice, the easier it will be
to figure out efficient techniques If a solution eludes you at first, sleep on it!
Try again the next day. Don’t give up! You will succeed!
AcknowledgementsThis presentation was made possible by
training and equipment from a Merced College Access to Technology grant.
Thank you to Marguerite Smith for the template for some of the slides.