Day 55 Verifying Identities 5.1 & 5.2

1. Day 55Verifying Identities5.1 & 5.2

2. What you will learn… • Recognize and write the fundamental trigonometric identities. • Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.

3. Plan for the Day • What are we doing in this chapter? • Review of identities from Chapter 4 • Techniques for verifying identities • Homework

4. Solving Non Trigonometric Equations Solve for x: 1. x + 3x – 4 = 2x – 7 2. (x + 1)2 – 3 = 4x + 1 Find the zeros of the function: 3. y = 3x +7 4. y = x2 + 7x + 12 What did you do to solve the problem?

5. Solving Trigonometric Functions Our ultimate goal is to solve trigonometric functions. To do so we have to be able to … • Simplify • Combine “like” terms (may need interpretation) • Factor

6. Another Piece We will also be verifying that expressions are equivalent or proving that a statement is an identity What is an identity?

7. Simplifying Equations and Verifying Identities To be able to solve many trigonometric problems you must first simplify the expression. Verifying identities means to demonstrate that two expressions represent the same thing. This allows you to replace one expression with another to help in simplifying. Verifying and simplifying uses the same set of skills and techniques.

8. Simplifying Equations and Verifying Identities Simplifying: How simple is simple enough? • No denominators • All like terms combined • All common factors have been eliminated

9. Which is more simple? Here are some equivalent terms; which is more simple A or B? • 5 B. 2 + 3 A. 6/2 B. 3 • 5/100 B. 1/20 A. 20 B. 4 • 5

10. Which is more simple? Here are some equivalent terms; which is more simple A or B? • 1 B. sin2θ + cos2θ A. sin θ/cosθ B. tan θ • 1/sec θ B. cosθ A. 1+ tan2 θ B. sec2 θ

11. What are Identities? What are some of the identities we have studied so far…? • Reciprocal • Co Function • Quotient • Pythagorean • Even / Odd

12. Reciprocal Functions sin  = 1/csc csc = 1/sin cos = 1/sec sec = 1/cos tan = 1/cot cot = 1/tan

13. Cofunctions sin  = cos(90  ) cos  = sin(90  ) tan  = cot(90  ) cot  = tan(90  ) sec  = csc(90  ) csc  = sec(90  )

14. Quotient and Pythagorean Identities Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2 cot2  + 1 = csc2 

15. Even and Odd Trig Functions • Cosine and secant functions are even cos (-t) = cos t sec (-t) = sec t • Sine, cosecant, tangent and cotangent are odd sin (-t) = - sin t csc (-t) = - csc t tan (-t) = - tan t cot (-t) = - cot t

16. Using Identities we know Adjustments to known identities: sin2  + cos2  = 1 can also be written as: sin2  = 1 – cos2  OR cos2  = 1 – sin2  The other Pythagorean Identities can be similarly adjusted.

17. Verifying Identities using Technology Checking with a calculator

18. Simplifying or Verifying Algebraically: How? There are several techniques or methods of approaching these problems based upon the structure of the problem…

19. Simplifying or Verifying Algebraically: How? • Substituting identities to eliminate like terms and simplify • Splitting rational functions so it is easier to see what identities can be used to simplify • Factoring • For rational functions, finding a common denominator to help simplify • Eliminating the denominator (creating binomials that can be simplified into a single term) • Combinations of all the above

20. Keep in Mind! • These problems take practice to get good at them! Even if you are stumped, try something! Even a path that leads to a dead end can provide valuable insight.

21. Using the Identities to Simplify • sec x cos x • tan2 x – sec2x • sin (-x) /cos (-x)

22. Try These

23. Splitting Rational Functions …then use the identities • cot (x) / csc (x) • sec θ • sin θ / tan θ • (1 + sin θ) / cos θ

25. Factoring Before moving on to factoring expressions that have trigonometric functions, let’s review factoring linear and quadratic functions.

26. Factor Simple factoring: take out common factor 1. x2 - xy 2. 2x – 4xy Factoring quadratics 3. x2 - 2x + 1 4. x2 - 3

27. Factoring with Trig Functions • Simpletan2 x – tan2 x sin2 x Now that it is factored, can it be simplified using identities 2. Trinomial tan4 x + 2 tan2 x + 1 Now that it is factored, can it be simplified using identities

28. Try these

29. Combination Multiple Manipulations 2 sec2 x – 2 sec2 x sin2 x - sin2 x – cos2 x

30. Simplifying vs Verifying • Simplifying is taking an expression that must be worked until it is in the simplest form: • No denominators • Combining like terms • Eliminating common factors • Verifying is taking an equation and make one side of the equation look like the other. For our activities we will work only one side of the equation. Never divide by a variable.

31. More next time…

32. Homework 29 • 5.1 Page 359 15-26 all (matching – helps you see where you are going.) • 5.2 page 367 1-13 odd, 27 – 30 all (work one side of the equation only.)