SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)

Transcription

1 SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e

2 2 P a g e

3 Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians. - Verify, with or without technology, that a given value is a solution to a trigonometric equation. - Determine, algebraically, the solution of a trigonometric equation, stating the solution in exact form when possible. - Determine, using technology, the approximate solution of a trigonometric equation in a restricted domain - Relate the general solution of a trigonometric equation to the zeros of the corresponding trigonometric function (restricted to sine and cosine functions). - Determine, using technology, the general solution of a given trigonometric equation - Identify and correct errors in a solution for a trigonometric equation. Key ideas: Verify means to see if it works for a certain value of the variable. If you put in a value and its false then it cannot be an identity. If you put in a value and it is true then it may be an identity. You may also verify a trig identity graphically in your calculator (both graphs should appear the same). Verifying an equation is not enough to conclude that the equation is an identity. Prove: means to show that the left side is equal to the right side for all values of the variable in the domain. When proving you may not move anything from one side of the equation to the other. You have to work with each side of the equation independently. - Try to simplify the more complicated side - Use identities from your formula sheet to make substitutions - Rewrite in terms of sine and cosine only - factor Solve: find the particular value of the variable for which the equation is true. When solving you may do the same thing to both sides of the equation to isolate the variable. - Isolate trig ratio - Factor with one side of the equation being equal to zero - Simplify using trig identities then solve - Remember to check for non-permissible values. 3 P a g e

4 7 Example 1: Verify that is a solution to the trigonometric equation 6 3 5sin 2 1 3sin in the domain. 2 Example 2: Determine the exact roots for the trigonometric equation 2cos 3 0 in the domain 0 2. Example 3: Solve the trigonometric equation in the specified domain. 2sec 6 0, Example 4: How is the general solution of 2 y 2cos 3cos 1? 2 2cos 3cos 1 0 related to the zeros of 4 P a g e

5 Example 5: Determine the general solution for 3csc 6 0. Example 6: Solve the following trigonometric equation. 2 9sin 12sin 4 0, 0, 360 Practice Questions: 1. 5 P a g e

6 2. 6 P a g e

7 P a g e

8 P a g e

9 P a g e

10 Specific Outcome 6: Prove trigonometric identities, using: reciprocal identities quotient identities Pythagorean identities sum or difference identities (restricted to sine, cosine and tangent) double-angle identities (restricted to sine, cosine and tangent). - Explain the difference between a trigonometric identity and a trigonometric equation - Verify a trigonometric identity numerically for a given value in either degrees or radians. - Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid. - Determine, graphically, the potential validity of a trigonometric identity, using technology. - Determine the non-permissible values of a trigonometric identity - Prove, algebraically, that a trigonometric identity is valid. - Determine, using the sum, difference and double-angle identities, the exact value of a trigonometric ratio. Key ideas: trigonometric equation An equation involving trigonometric ratios. trigonometric identity A trigonometric equation that is true for all permissible values of the variable in the expression on both sides of the equation. Note: Sometimes the symbol is used instead of = to indicate that a statement is an identity. 10 P a g e

11 Examples: Example 7: Verify that the equation 30 and for. 4 sec sin tan cot is true for Example 8: Explain why verifying that the two sides of a trigonometric identity are equal for given values is insufficient to conclude that the identity is valid. sin cos 1 cos Example 9: Consider the equation. Graph the two sides of the 1 cos tan equation using technology, over the domain 0 2. Could it be an identity? 11 P a g e

12 1 cos sin Example 10: Consider the identity. What are the non-permissible sin 1 cos values for the identity over the domain Example 11: Prove each identity. a) cos cot cos cot sec tan b) cos sec tan 1 sin 12 P a g e

13 c) Prove the identity sin2x= 2sinx cosx Example 12: Determine the exact value for each expression. a) sin 12 b) cos75 13 P a g e

14 Practice: P a g e

15 P a g e

16 P a g e

17 P a g e

18 P a g e

19 Practice Questions Answers 1. C 2. D 3. D 4. A 5. B 6. C 7. A 8. B 9. C 10. A 11. D 12. A 13. C 14. A 15. C 16. D 17. D 18. A 19. C 20. C N.R P a g e