Verifying Trigonometric Identities

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Verifying Trigonometric Identities
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What is an Identity?

• An identity is a statement that two expressions
  are equal for every value of the variable.
• Examples

The left-hand expression always equals the
right-hand expression, no matter what x equals.

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The fundamental Identities

• Reciprocal Identities

• Quotient Identities

The beauty of the identities is that we can get
all functions in terms of sine and cosine.
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The Fundamental Identities

• Identities for Negatives

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The Fundamental Identities

• Pythagorean Identities

X
The only unique Identity here is the top one, the
other two can be obtained using the top identity.
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Variations of Identities using Arithmetic

• Variations of these Identities

We can create different versions of many of these
identities by using arithmetic.
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Lets look at some examples!
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Verifying Trigonometric Identities

• Now we continue on our journey!

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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.

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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.

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Example
and
Since both sides are the same, the identity is
verified.
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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 youre really stuck make sure to

Change everything on both sides to sine and
cosine.
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Remember to

• Work with only one side at a time!

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Establish the following identity
Let's sub in here using reciprocal identity
We are done! We've shown the LHS equals the RHS
We often use the Pythagorean Identities solved
for either sin2? or cos2?. sin2? cos2? 1
solved for sin2? is sin2? 1 - cos2? which is
our left-hand side so we can substitute.
In establishing an identity you should NOT move
things from one side of the equal sign to the
other. Instead substitute using identities you
know and simplifying on one side or the other
side or both until both sides match.
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Establish the following identity
Let's sub in here using reciprocal identity and
quotient identity
We worked on LHS and then RHS but never moved
things across the sign
FOIL denominator
combine fractions
Another trick if the denominator is two terms
with one term a 1 and the other a sine or cosine,
multiply top and bottom of the fraction by the
conjugate and then you'll be able to use the
Pythagorean Identity on the bottom
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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!

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Dont 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. Dont give up!
• You will succeed!

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Establish the identity
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Establish the identity
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Establish the identity
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Homework

• 14.3 pg 780 s 25-28 all, 29-35 odd

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Acknowledgements

• This 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.