2. What is an Identity? <ul><li>An identity is a statement that two expressions are equal for every value of the variable. </li></ul><ul><li>Examples: </li></ul>The left-hand expression always equals the right-hand expression, no matter what x equals.
3. The fundamental Identities <ul><li>Reciprocal Identities </li></ul><ul><li>Quotient Identities </li></ul>The beauty of the identities is that we can get all functions in terms of sine and cosine.
4. The Fundamental Identities <ul><li>Identities for Negatives </li></ul>
5. The Fundamental Identities <ul><li>Pythagorean Identities </li></ul>The only unique Identity here is the top one, the other two can be obtained using the top identity. X
6. Variations of Identities using Arithmetic <ul><li>Variations of these Identities </li></ul>We can create different versions of many of these identities by using arithmetic.
7. Let’s look at some examples!
8. Verifying Trigonometric Identities Now we continue on our journey!
9. An Identity is Not a Conditional Equation <ul><li>Conditional equations are true only for some values of the variable. </li></ul><ul><li>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. </li></ul><ul><li>We are not “solving” identities so we must approach identities differently. </li></ul>
10. We Verify (or Prove) Identities by doing the following: <ul><li>Work with one side at a time. </li></ul><ul><li>We want both sides to be exactly the same. </li></ul><ul><li>Start with either side </li></ul><ul><li>Use algebraic manipulations and/or the basic trigonometric identities until you have the same expression as on the other side. </li></ul>
11. Example: and Since both sides are the same, the identity is verified.
12. Suggestions <ul><li>Start with the more complicated side </li></ul><ul><li>Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier) </li></ul><ul><li>Try algebra: factor, multiply, add, simplify, split up fractions </li></ul><ul><li>If you’re really stuck make sure to: </li></ul>Change everything on both sides to sine and cosine.
13. Remember to: <ul><li>Work with only one side at a time! </li></ul>
14. Establish the following identity: 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. Let's sub in here using reciprocal identity We often use the Pythagorean Identities solved for either sin 2 or cos 2 . sin 2 + cos 2 = 1 solved for sin 2 is sin 2 = 1 - cos 2 which is our left-hand side so we can substitute. We are done! We've shown the LHS equals the RHS
15. Establish the following identity: Let's sub in here using reciprocal identity and quotient identity 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 We worked on LHS and then RHS but never moved things across the = sign combine fractions FOIL denominator
16. How to get proficient at verifying identities: <ul><li>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. </li></ul><ul><li>Redo the examples done in class using the same approach, this will help you build confidence in your instincts! </li></ul>
17. Don’t Get Discouraged! <ul><li>Every identity is different </li></ul><ul><li>Keep trying different approaches </li></ul><ul><li>The more you practice, the easier it will be to figure out efficient techniques </li></ul><ul><li>If a solution eludes you at first, sleep on it! Try again the next day. Don’t give up! </li></ul><ul><li>You will succeed! </li></ul>
23. Acknowledgements <ul><li>This presentation was made possible by training and equipment from a Merced College Access to Technology grant. </li></ul><ul><li>Thank you to Marguerite Smith for the template for some of the slides. </li></ul>