1. Using Fundamental Identities MATH 109 - Precalculus S. Rook

2. Overview Section 5.1 in the textbook: – Using identities to evaluate a function – Simplifying trigonometric expressions – Factoring trigonometric expressions – Adding & subtracting trigonometric expressions – Eliminating fractions in a trigonometric expression 2

3. Using Identities to Evaluate a Function

4. Review of Identities Discussed thus Far Examine the chart on page 374 – you must be familiar with these identities: – Reciprocal – Quotient/Ratio – Pythagorean – Cofunction – Even/Odd 4

5. Using Identities to Evaluate a Function Given the value of a trigonometric function, sometimes we wish to deduce the value of another trigonometric function: – Try to get the values in terms of sines & cosines 5

6. Using Identities to Evaluate a Function (Example) Ex 1: Use the given values to evaluate (if possible) all six trigonometric functions: a), b) 6

7. Simplifying Trigonometric Expressions

8. There are times when we wish to write a trigonometric expression in an alternate form: – Identities are the key As expected, there is usually more than one way to simplify a trigonometric expression Some tips when simplifying: – Check to see if anything can be factored out – Express trigonometric functions in terms of sines & cosines 8

9. Simplifying Trigonometric Expressions (Example) Ex 2: Simplify using the fundamental identities – possible to have multiple correct answers: a) b) 9

10. Simplifying Trigonometric Expressions (Example) Ex 3: Use the trigonometric substitution to write the algebraic expression as a trigonometric function of θ, 0 < θ < π ⁄ 2 10

11. Factoring Trigonometric Expressions

12. Recall our strategies for factoring a quadratic: – i.e. easy & hard trinomials, difference of two squares, etc Can apply the same strategies when trigonometric functions are involved: – E.g. – Permissible to substitute Just remember to revert back to the trigonometric function at the end E.g. Let x = csc θ: 12

13. Factoring Trigonometric Expressions (Continued) We must have the SAME trigonometric function in an expression In order to factor – Pythagorean identities are helpful in this situation – e.g. cannot be factored initially 13

14. Factoring Trigonometric Expressions (Example) Ex 4: Factor the trigonometric expression and then simplify if possible: a) b) 14

15. Adding & Subtracting Trigonometric Expressions

16. Sometimes we wish to condense multiple sums & differences of fractions containing trigonometric functions into one fraction – Find an LCD e.g. LCD of is (sin x)(cos x) e.g. LCD of is (sec x)(sec x + 1) e.g. LCD of is tan 2 θ 16

17. Adding & Subtracting Trigonometric Expressions (Example) Ex 5: Simplify by combining over one fraction: a) b) 17

18. Eliminating Fractions in a Trigonometric Expression

19. An important skill to master especially for Calculus is to be able to rewrite a trigonometric expression without a fraction The main idea is to get ONE term in the denominator – We can then divide all terms of the numerator by the denominator Only works when there is ONE term in the denominator If the denominator contains two terms: – Multiply by the conjugate of the denominator Results in a difference of two squares – Apply a Pythagorean identity to condense to one term 19

20. Eliminating Fractions in a Trigonometric Expression (Example) Ex 6: Rewrite the expression so that it is not in fractional form: a) b) 20

21. Summary After studying these slides, you should be able to: – Utilize the identities on page 474 – Use identities to evaluate trigonometric functions – Simplify trigonometric expressions – Factor trigonometric expressions – Add & subtract trigonometric expressions – Eliminate fractions in a trigonometric expression Additional Practice – See the list of suggested problems for 5.1 Next lesson – Verifying Trigonometric Identities (Section 5.2) 21