Trigonometric Equations: Challenges and Solutions

1. Get out paper for notes!!!

2. Warm-up (3:30 m) • Solve for all solutions graphically: sin3x = –cos2x • Molly found that the solutions to cos x = 1 are x = 0 + 2kπ AND x = 6.283 + 2kπ, . Is Molly’s solution correct? Why or why not?

3. sin3x = –cos2x

4. cos x = 1 • x = 0 + 2kπ • x = 6.283 + 2kπ,

5. Solving Trigonometric Equations Algebraically

6. Inverse Trigonometric Functions • Remember, your calculator must be in RADIAN mode. • cos x = 0.6 • We can use inverse trig functions to solve for x.

7. Check the solution graphically

8. Why are there two solutions? Let’s consider the Unit Circle Where is x (cosine) positive?

9. “All Students Take Calculus” S A sine is positive all ratios are positive cosecant is positive T C cosine is positive tangent is positive cotangent is positive secant is positive

10. How do we find the other solutions algebraically?

11. cos x = 0.6

12. Your Turn: • Solve for all solutions algebraically: cos x = – 0.3

13. sin x = –0.75

14. Your Turn: • Solve for all solutions algebraically: sin x = 0.5

15. What about tangent? • The solution that you get in the calculator is the only one! tan x = –5

16. Your Turn: • Solve for all solutions algebraically: 1. cos x = –0.2 2. sin x = – ⅓ 3. tan x = 3 4. sin x = 4

17. What’s going on with #4? • sin x = 4

18. How would you solve for x if… 3x2 – x = 2

19. So what if we have… 3 sin2x – sin x = 2

20. What about… tan x cos2x – tan x = 0

21. Your Turn: • Solve for all solutions algebraically: 5. 4 sin2x = 5 sin x – 1 6. cos x sin2x = cos x 7. sin x tan x = sin x 8. 5 cos2x + 6 cos x = 8

22. Warm-up (4 m) 1. Solve for all solutions algebraically: 3 sin2x + 2 sin x = 5 • Explain why we would reject the solution cos x = 10

23. 3 sin2x + 2 sin x = 5

24. Explain why we would reject the solution cos x = 10

25. What happens if you can’t factor the equation? Quadratic Formula • x2 + 5x + 3 = 0 The plus or minus symbol means that you actually have TWO equations!

26. x2 + 5x + 3 = 0ax2 + bx + c = 0

27. Using the Quadratic Equation to Solve Trigonometric Equations • You can’t mix trigonometric functions. (Only one trigonometric function at a time!) • Must still follow the same basic format: • ax2 + bx + c = 0 • 2 cos2x + 6 cos x – 4 = 0 • 7 tan2x + 10 = 0

28. tan2x + 5 tan x + 3 = 0

29. 3 sin2x – 8 sin x = –3

30. Your Turn: • Solve for all solutions algebraically: • sin2x + 2 sin x – 2 = 0 • tan2x – 2 tan x = 2 • cos2x = –5 cos x + 1

31. Warm-up (4 m) • Solve for all solutions algebraically: • tan x cos x + 3 tan x = 0 • 2 cos2x + 7 cos x – 1 = 0

32. tan x cos x + 3 tan x = 0

33. 2 cos2x + 7 cos x – 1 = 0

34. Seek and Solve! You have 30 m to complete the seek and solve. Show all your work on a sheet of paper because I’m collecting it for a classwork grade.

35. Remember me?

36. Using Reciprocal Identities to Solve Trigonometric Equations • Our calculators don’t have reciprocal function (sec x, csc x, cot x) keys. • We can use the reciprocal identities to rewrite secant, cosecant, and cotangent in terms of cosine, sine, and tangent!

38. cot x cos x = cos x

39. Your Turn: • Use the reciprocal identities to solve for solutions algebraically: 1. cot x = –10 2. tan x sec x + 3 tan x = 0 3. cos x csc x = 2 cos x

40. Using Pythagorean Identities to Solve Trigonometric Equations • You can use a Pythagorean identity to solve a trigonometric equation when: • One of the trig functions is squared • You can’t factor out a GCF • Using a Pythagorean identity helps you rewrite the squared trig function in terms of the other trig function in the equation

41. cos2x – sin2x + sin x = 0

42. sec2x – 2 tan2x = 0

43. sec2x + tan x = 3

44. Your Turn: • Use Pythagorean identities to solve for all solutions algebraically: • –10 cos2x – 3 sin x + 9 = 0 • –6 sin2x + cos x + 5 = 0 • sec2x + 5 tan x = –2