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Verifying Trig Identities (5.1)

Verifying Trig Identities (5.1). JMerrill, 2009 (contributions from DDillon). 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

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Verifying Trig Identities (5.1)

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  1. Verifying Trig Identities(5.1) JMerrill, 2009 (contributions from DDillon)

  2. 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

  3. Recall

  4. Recall - Identities Reciprocal Identities Also true:

  5. Recall - Identities Quotient Identities

  6. Fundamental Trigonometric Identities Negative Identities (even/odd) These are the only even functions!

  7. Recall - Identities Cofunction Identities

  8. Recall - Identities Pythagorean Identities

  9. 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

  10. Simplifying # 1

  11. Simplifying #2

  12. Simplifying #3

  13. Simplifying #4

  14. Simplifying #5

  15. 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 don’t work. ALWAYS TRY SOMETHING!!!

  16. Example • Prove • 2 fractions that need to be added: • Shortcut:

  17. 1 + cot2x = csc2 x

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