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6.5 Basic Trigonometric Identities

6.5 Basic Trigonometric Identities. Objective: Develop basic trigonometric identities. Trigonometric Identities. Trigonometric identities are equations that are true for all values of the variable for which the equation is defined. They are most often used to simplify an expression.

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6.5 Basic Trigonometric Identities

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  1. 6.5 Basic Trigonometric Identities Objective: Develop basic trigonometric identities.

  2. Trigonometric Identities • Trigonometric identities are equations that are true for all values of the variable for which the equation is defined. They are most often used to simplify an expression. • Algebraic rules are the same for trigonometric expressions, the notation is sometimes just slightly different.

  3. Quotient Identities • In Chapter 6 we introduced the Unit Circle & we also learned our first identities:

  4. Reciprocal Identities • We also learned the reciprocal identities.

  5. Example #1 • Simplify the following expressions: A) B)

  6. Example #2

  7. Pythagorean Identities • Using the Pythagorean Theorem & the Unit Circle, the Pythagorean Identities are created:

  8. Pythagorean Identities continued… Be very careful when using Pythagorean identities, the expressions must be squared:

  9. Example #3 • Simplify the following expressions. A) This problem utilizes two identities:

  10. Example #3 • Simplify the following expressions. B) Two identities are also used here:

  11. Example #4 • Use the Quotient, Reciprocal, and Pythagorean identities to find the remaining 5 trigonometric functions.

  12. Periodicity Identities

  13. Periodicity Identities

  14. Negative Angle Identities

  15. Example #5 • Simplify the following expression. This problem required first factoring the top and using the identity: After the substitution, the bottom was factored and the top rearranged. Canceling like terms gives the answer shown.

  16. Example #6 • Simplify the following expression. Here the expression has a GCF factored out first. Then a substitution is made with this identity: Then the reciprocal identity is used and like terms are canceled. Finally, the reciprocal identity is used again.

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