Verifying Trig Identities (5.1)

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Verifying Trig Identities(5.1)

• JMerrill, 2009
• (contributions from DDillon)

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Trig Identities

• Identity an equation that is true for all values
  of the variable for which the expressions are
  defined
• Ex or (x 2) x 2
• Conditional Equation only true for some of the
  values
• Ex tan x 0 or x2 3x 2 0

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Recall
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Recall - Identities
Reciprocal Identities
Also true
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Recall - Identities
Quotient Identities
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Fundamental Trigonometric Identities
Negative Identities (even/odd)
These are the only even functions!
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Recall - Identities
Cofunction Identities
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Recall - Identities
Pythagorean Identities
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Simplifying Trig Expressions

• Strategies
• Change all functions to sine and cosine (or at
  least into the same function)
• Substitute using Pythagorean Identities
• Combine terms into a single fraction with a
  common denominator
• Split up one term into 2 fractions
• Multiply by a trig expression equal to 1
• Factor out a common factor

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Simplifying 1
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Simplifying 2
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Simplifying 3
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Simplifying 4
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Simplifying 5
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Proof Strategies

• Never cross over the equal sign (you cannot
  assume equality)
• Transform the more complicated side of the
  identity into the simpler side.
• Substitute using Pythagorean identities.
• Look for opportunities to factor
• Combine terms into a single fraction with a
  common denominator, or split up a single term
  into 2 different fractions
• Multiply by a trig expression equal to 1.
• Change all functions to sines and cosines, if the
  above ideas dont work.

ALWAYS TRY SOMETHING!!!
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Example

• Prove
• 2 fractions that need to be added
• Shortcut

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1 cot2x csc2 x
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