1. 5.3 S OLVING T RIG EQUATIONS

2. S OLVING T RIG E QUATIONS Solve the following equation for x: Sin x = ½

3. S OLVING T RIG E QUATIONS In this section, we will be solving various types of trig equations You will need to use all the procedures learned last year in Algebra II All of your answers should be angles. Note the difference between finding all solutions and finding all solutions in the domain [0, 2π)

4. S OLVING T RIG E QUATIONS Guidelines to solving trig equations: 1) Isolate the trig function 2) Find the reference angle 3) Put the reference angle in the proper quadrant(s) 4) Create a formula for all possible answers (if necessary)

5. S OLVING T RIG E QUATIONS 1- 2 Cos x = 0 1) Isolate the trig function 1- 2 Cos x = 0 + 2 Cos x = + 2 Cos x 1= 2 Cos x 2 Cos x = ½

6. S OLVING T RIG E QUATIONS Cos x = ½ 2) Find the reference angle x = 3) Put the reference angle in the proper quadrant(s) I =IV =

7. S OLVING T RIG E QUATIONS Cos x = ½ 4) Create a formula if necessary x =

8. S OLVING T RIG E QUATIONS Find all solutions to the following equation: Sin x + 1 = - Sin x + Sin x → 2 Sin x + 1 = 0 - 1 → 2 Sin x = -1 → Sin x = - ½

9. S OLVING T RIG E QUATIONS Sin x = - ½ Ref. Angle:Quad.: III: Iv:

10. S OLVING T RIG E QUATIONS Find the solutions in the interval [0, 2π) for the following equation: Tan²x – 3 = 0 Tan²x = 3 Tan x =

11. S OLVING T RIG E QUATIONS Tan x = Ref. Angle:Quad.: I: II: III:IV: x =

12. S OLVING T RIG E QUATIONS Solve the following equations for all real values of x. a) Sin x + = - Sin x b) 3Tan² x – 1 = 0 c) Cot x Cos² x = 2 Cot x

13. S OLVING T RIG E QUATIONS Find all solutions to the following equation: Sin x + = - Sin x 2 Sin x = - Sin x = - x =

14. S OLVING T RIG E QUATIONS 3Tan² x – 1 = 0 Tan² x = Tan x = x =

15. S OLVING T RIG E QUATIONS Cot x Cos² x = 2 Cot x Cot x Cos² x – 2 Cot x = 0 Cot x (Cos² x – 2) = 0 Cot x = 0 Cos² x – 2 = 0 Cos x = Cos x = 0 x = No Solution

16. S OLVING T RIG E QUATIONS

17. 5.3 S OLVING T RIG EQUATIONS

18. S OLVING T RIG E QUATIONS Find all solutions to the following equation. 4 Tan²x – 4 = 0 Tan²x = 1 Tan x = ±1 Ref. Angle = x =

19. S OLVING T RIG E QUATIONS Equations of the Quadratic Type Many trig equations are of the quadratic type: 2Sin²x – Sin x – 1 = 0 2Cos²x + 3Sin x – 3 = 0 To solve such equations, factor the quadratic or, if that is not possible, use the quadratic formula

20. S OLVING T RIG E QUATIONS Solve the following on the interval [0, 2π) 2Cos²x + Cos x – 1 = 0 If possible, factor the equation into two binomials. 2x² + x - 1 (2Cos x – 1) (Cos x + 1) = 0 Now set each factor equal to zero

21. S OLVING T RIG E QUATIONS 2Cos x – 1 = 0 Cos x + 1 = 0 Cos x = ½ Ref. Angle: Quad:I, IV x = Cos x = -1 x =

22. S OLVING T RIG E QUATIONS Solve the following on the interval [0, 2π) 2Sin²x - Sin x – 1 = 0 (2Sin x + 1) (Sin x - 1) = 0

23. S OLVING T RIG E QUATIONS 2Sin x + 1 = 0 Sin x - 1 = 0 Sin x = - ½ Ref. Angle: Quad:III, IV x = Sin x = 1 x =

24. S OLVING T RIG E QUATIONS Solve the following on the interval [0, 2π) 2Cos²x + 3Sin x – 3 = 0 Convert all expressions to one trig function 2 (1 – Sin²x) + 3Sin x – 3 = 0 2 – 2Sin²x + 3Sin x – 3 = 0 0 = 2Sin²x – 3Sin x + 1

25. S OLVING T RIG E QUATIONS 2Sin x - 1 = 0 Sin x - 1 = 0 Sin x = ½ Ref. Angle: Quad:I, II x = Sin x = 1 x = 0 = 2Sin²x – 3Sin x + 10 = (2Sin x – 1) (Sin x – 1)

26. S OLVING T RIG E QUATIONS Solve the following on the interval [0, 2π) 2Sin²x + 3Cos x – 3 = 0 Convert all expressions to one trig function 2 (1 – Cos²x) + 3Cos x – 3 = 0 2 – 2Cos²x + 3Cos x – 3 = 0 0 = 2Cos²x – 3Cos x + 1

27. S OLVING T RIG E QUATIONS 2Cos x - 1 = 0 Cos x - 1 = 0 Cos x = ½ Ref. Angle: Quad:I, IV x = Cos x = 1 x = 0 = 2Cos²x – 3Cos x + 10 = (2Cos x – 1) (Cos x – 1)

28. S OLVING T RIG E QUATIONS The last type of quadratic equation would be a problem such as: Sec x + 1 = Tan x What do these two trig functions have in common? When you have two trig functions that are related through a Pythagorean Identity, you can square both sides. ( )² ²

29. S OLVING T RIG E QUATIONS (Sec x + 1)² = Tan²x Sec²x + 2Sec x + 1= Sec²x - 1 2 Sec x + 1 = -1 Sec x = -1 Cos x = -1 x = When you have a problem that requires you to square both sides, you must check your answer when you are done!

30. S OLVING T RIG E QUATIONS Sec x + 1 = Tan xx =

31. S OLVING T RIG E QUATIONS Cos x + 1 = Sin x Cos²x + 2Cos x + 1 = 1 – Cos² x 2Cos² x + 2 Cos x = 0 Cos x (2 Cos x + 2) = 0 Cos x = 0 x = (Cos x + 1)² = Sin² x Cos x = - 1 x =

32. S OLVING T RIG E QUATIONS Cos x + 1 = Sin x x =

33. 5.3 S OLVING T RIG EQUATIONS

34. S OLVING T RIG E QUATIONS Equations involving multiply angles Solve the equation for the angle as your normally would Then divide by the leading coefficient

35. S OLVING T RIG E QUATIONS Solve the following trig equation for all values of x. 2Sin 2x + 1 = 0 2Sin 2x = -1 Sin 2x = - ½ 2x = x =

36. S OLVING T RIG E QUATIONS Redundant Answer

37. S OLVING T RIG E QUATIONS Solve the following equations for all values of x. a) 2Cos 3x – 1 = 0 b) Cot (x/2) + 1 = 0

38. S OLVING T RIG E QUATIONS 2Cos 3x - 1 = 0 2Cos 3x = 1 Cos 3x = ½ 3x = x =

39. S OLVING T RIG E QUATIONS

40. Topics covered in this section: Solving basic trig equations Finding solutions in [0, 2π) Find all solutions Solving quadratic equations Squaring both sides and solving Solving multiple angle equations Using inverse functions to generate answers

41. S OLVING T RIG E QUATIONS Find all solutions to the following equation: Sec²x – 3Sec x – 10 = 0 (Sec x + 2) (Sec x – 5) = 0 Sec x + 2 = 0 Sec x – 5 = 0 Sec x = -2 Cos x = - ½ x = Sec x = 5 Cos x = x =

42. S OLVING T RIG E QUATIONS One of the following equations has solutions and the other two do not. Which equations do not have solutions. a) Sin²x – 5Sin x + 6 = 0 b) Sin²x – 4Sin x + 6 = 0 c) Sin²x – 5Sin x – 6 = 0 Find conditions involving constants b and c that will guarantee the equation Sin²x + bSin x + c = 0 has at least one solution.

43. S OLVING T RIG F UNCTIONS Find all solutions of the following equation in the interval [0, 2π) Sec²x – 2 Tan x = 4 1 + Tan²x – 2Tan x – 4 = 0 Tan²x – 2Tan x – 3 = 0 (Tan x + 1) (Tan x – 3) = 0 Tan x = -1 Tan x = 3

44. S OLVING T RIG F UNCTIONS Tan x = -1Tan x = 3 x = ArcTan 3 ref. angle: 71.6º Quad: I, III x = 71.6º, 251.6º