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5.3 Solving Trig equations

5.3 Solving Trig equations. Solving Trig Equations. Solve the following equation for x: Sin x = ½. Solving Trig Equations. 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

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5.3 Solving Trig equations

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  1. 5.3 Solving Trig equations

  2. Solving Trig Equations • Solve the following equation for x: Sin x = ½

  3. Solving Trig Equations • 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. Solving Trig Equations • Guidelines to solving trig equations: • Isolate the trig function • Find the reference angle • Put the reference angle in the proper quadrant(s) • Create a formula for all possible answers (if necessary)

  5. Solving Trig Equations 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 2 Cos x = ½

  6. Solving Trig Equations Cos x = ½ 2) Find the reference angle x = 3) Put the reference angle in the proper quadrant(s) I = IV =

  7. Solving Trig Equations Cos x = ½ 4) Create a formula if necessary x = x =

  8. Solving Trig Equations • Find all solutions to the following equation: Sin x + 1 = - Sin x + Sin x + Sin x → 2 Sin x + 1 = 0 - 1 - 1 → 2 Sin x = -1 → Sin x = - ½

  9. Solving Trig Equations Sin x = - ½ Ref. Angle: Quad.: III: Iv:

  10. Solving Trig Equations • Find the solutions in the interval [0, 2π) for the following equation: Tan²x – 3 = 0 Tan²x = 3 Tan x =

  11. Solving Trig Equations Tan x = Ref. Angle: Quad.: I: III: IV: II: x =

  12. Solving Trig Equations • Solve the following equations for all real values of x. • Sin x + = - Sin x • 3Tan² x – 1 = 0 • Cot x Cos² x = 2 Cot x

  13. Solving Trig Equations • Find all solutions to the following equation: Sin x + = - Sin x 2 Sin x = - x = Sin x = - x =

  14. Solving Trig Equations 3Tan² x – 1 = 0 x = Tan² x = x = Tan x = x = x =

  15. Solving Trig Equations 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 = 0 Cos² x – 2 = 0 x = Cos x = No Solution x =

  16. Solving Trig Equations

  17. 5.3 Solving Trig equations

  18. Solving Trig Equations • Find all solutions to the following equation. 4 Tan²x – 4 = 0 x = Tan²x = 1 x = Tan x = ±1 Ref. Angle =

  19. Solving Trig Equations • 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. Solving Trig Equations • Solve the following on the interval [0, 2π) 2Cos²x + Cos x – 1 = 0 2x² + x - 1 If possible, factor the equation into two binomials. (2Cos x – 1) (Cos x + 1) = 0 Now set each factor equal to zero

  21. Solving Trig Equations 2Cos x – 1 = 0 Cos x + 1 = 0 Cos x = ½ Cos x = -1 Ref. Angle: x = Quad: I, IV x =

  22. Solving Trig Equations • Solve the following on the interval [0, 2π) 2Sin²x - Sin x – 1 = 0 (2Sin x + 1) (Sin x - 1) = 0

  23. Solving Trig Equations 2Sin x + 1 = 0 Sin x - 1 = 0 Sin x = - ½ Sin x = 1 Ref. Angle: x = Quad: III, IV x =

  24. Solving Trig Equations • 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. Solving Trig Equations 0 = 2Sin²x – 3Sin x + 1 0 = (2Sin x – 1) (Sin x – 1) 2Sin x - 1 = 0 Sin x - 1 = 0 Sin x = ½ Sin x = 1 Ref. Angle: x = Quad: I, II x =

  26. Solving Trig Equations • 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. Solving Trig Equations 0 = 2Cos²x – 3Cos x + 1 0 = (2Cos x – 1) (Cos x – 1) 2Cos x - 1 = 0 Cos x - 1 = 0 Cos x = ½ Cos x = 1 Ref. Angle: x = Quad: I, IV x =

  28. Solving Trig Equations • 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. Solving Trig Equations (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. Solving Trig Equations Sec x + 1 = Tan x x =

  31. Solving Trig Equations (Cos x + 1)² = Sin² x 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 Cos x = - 1 x = x =

  32. Solving Trig Equations Cos x + 1 = Sin x x =

  33. 5.3 Solving Trig equations

  34. Solving Trig Equations • Equations involving multiply angles • Solve the equation for the angle as your normally would • Then divide by the leading coefficient

  35. Solving Trig Equations • Solve the following trig equation for all values of x. 2Sin 2x + 1 = 0 2Sin 2x = -1 Sin 2x = - ½ 2x = 2x = x = x =

  36. Solving Trig Equations Redundant Answer

  37. Solving Trig Equations • Solve the following equations for all values of x. • 2Cos 3x – 1 = 0 • Cot (x/2) + 1 = 0

  38. Solving Trig Equations 2Cos 3x - 1 = 0 2Cos 3x = 1 Cos 3x = ½ 3x = 3x = x = x =

  39. Solving Trig Equations

  40. Solving Trig Equations • 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. Solving Trig Equations 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 = 5 Sec x = -2 Cos x = Cos x = - ½ x = x =

  42. Solving Trig Equations • One of the following equations has solutions and the other two do not. Which equations do not have solutions. • Sin²x – 5Sin x + 6 = 0 • Sin²x – 4Sin x + 6 = 0 • 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. Solving Trig Functions • 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 = 3 Tan x = -1

  44. Solving Trig Functions Tan x = -1 Tan x = 3 x = ArcTan 3 ref. angle: 71.6º I, III Quad: x = 71.6º, 251.6º

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