Section 7.2 Notes

1. Section 7.2 Notes Verifying Trigonometric Identities

2. 7.2 Notes A trigonometric identity is an equation that is true for all values of the variable in the equation.

3. 7.2 Notes

4. 7.2 Notes There are eleven basic trigonometric identities. You are familiar with most of them. The first six are called reciprocal identities:

5. 7.2 Notes From today’s “do now,” the point P can be expressed as

6. 7.2 Notes The next two basic trigonometric identities are called quotient identities:

7. 7.2 Notes From today’s “do now,” the point P can be expressed as

8. 7.2 Notes The last three basic trigonometric identities are called Pythagorean identities:

9. 7.2 Notes The last three basic trigonometric identities are called Pythagorean identities:

10. 7.2 Notes To verify an identity means to show that an equation is true by simplifying one side of the equation until the equation is the same on both sides of the equal sign. In this lesson, you will learn how to verify identities. Here’s an identity that you will see verified:

11. Several lines of work will follow the original equation above, then you will arrive at a line of work like the one below. When both sides of the equation are the same, you have verified the identity.

12. 7.2 Notes Here are four strategies for verifying identities: • Work on one side of the equation only. • Substitute one or more basic trigonometric identities to simplify the expression on one side of the equation. • Factor, multiply, and/or cancel to simplify the expression on one side of the equation. • Multiply by a strategic form of one.

13. 7.2 Notes Let’s look at strategy 1 in more detail… 1. Work on one side of the equation only: *Start on the more complicated looking side: • Usually, we consider the complicated side to be the side with more trigonometric functions. • Sometimes it is the side with the fewest sine and/or cosine functions. • Sometimes it is the side with addition and/or subtraction of trigonometric functions.

14. 7.2 Notes Note: If you get stuck (can’t further simplify) on the side you chose to start on, you may work on the other side until you get unstuck. But, then you must go back and finish on the same side you began. Example 1Example 2

15. 7.2 Notes Example 1: One person may consider the left-hand side of the equation the more complicated side due to the tan2xterm. Someone else may consider the right-hand side of the equation the more complicated side due to the subtraction. This example is going to be worked on the left side, but it could be verified by working on the right side.

16. 7.2 Notes If both sides of the quotient identity are squared, the following identity will result:

17. 7.2 Notes Watch as is substituted for

18. Remember that anything divided by one is itself, so any number, variable, or trigonometric term can be thought of as being over one. Let’s put cos2x over 1:

19. The cos2x terms cancel out, leaving sin2x over 1 or simply sin2x:

20. The screen is starting to get full, so you’ll just see that last line of work for now. Soon you’ll see the entire identity. If the Pythagorean identity, is solved for sin2x you’ll get: Finish the identity with another substitution; replace sin2x with 1 – cos2x:

21. Watch as the same identify is verified again:

22. Notice the alignment of the equal marks. Copy this identity into your notes.

23. 7.2 Notes • Click here to see this identity verified again. • Ask Mr. Armistead if you have any questions about this example. • Click here if you would like to view the strategies for verifying identities before seeing a second example. • Click here to see a second example.

24. 7.2 Notes Example 2: Neither side of this equation has any sine or cosine terms; therefore, one may think that both sides are equally complicated. The right-hand side is probably considered more complicated due to the addition of terms as opposed to the multiplication of terms on the left side. This example is going to be worked on the right-hand side.

25. Begin by substituting both the cotA and the tanA terms with their respective quotient identities.

26. The right-hand side of the equation is the sum of fractions. Click here for a review of adding fractions. The common denominator of sinA and cosA is the product sinAcosA.

27. Compute the new numerator:

28. Using the Pythagorean identity, cos2A + sin2A=1, replace cos2A + sin2A with 1. Write the right-hand side as the product:

29. Make substitutions using the reciprocal identities for cosecant and secant: Watch the solution again and copy it into your notes making sure to align the equal marks.

30. Example 2:

31. 7.2 Notes Turn to p. 434 in your textbook and answer 5, 6, & 8. Use the following links: • Strategies for verifying identities • Example 1 • Example 2 • Adding Fractions Review • Solutions to p. 434: 5, 6, 8

32. p. 434: #5

33. p. 434: #6

34. p. 434: #8

35. 7.2 Notes Adding Fractions Review If two fractions have the same denominator, add their numerators and keep the denominator. Simplify when possible.

36. 7.2 Notes Adding Fractions Review In this example, the two fractions do not have the same denominator. Find the least common denominator. The least common denominator of 2 and 6 is 6. Find the new numerators. 2 divides into 6 three times, 3 times –1 is –3.

37. 7.2 Notes Adding Fractions Review And 6 divides into 6 one time, 1 times 5 is 5. Now add the numerators. Simplify.

38. 7.2 Notes Adding Fractions Review Here’s one more example of adding fractions that do not have the same denominator. Find the least common denominator. The common denominator of 3 and 4 is 12. Find the new numerators. 3 divides into 12 four times, 4 times 2 is 8.

39. 7.2 Notes Adding Fractions Review And 4 divides into 12 three times, 3 times 3 is 9. Now add the numerators. The sum is in simplest form. Click here to return to example 2.