Day 60 Solving Trigonometric Equations 5.3

1. Day 60Solving Trigonometric Equations 5.3

2. Objectives To find all solutions to trigonometric functions using the general form of the answer To find all solutions to trigonometric functions with multiple angles and partial angles

3. Plan for the Day Review of solving trigonometric equations over the interval [0, 2π) Writing solutions in the “general form” representing all solutions Solving Multiple and Partial Angles Homework: 5.3 Page 376 # 19, 20, 33-38 all

4. Solving Trigonometric Equations Purpose: Finding the angles that make the statement true. Several methods for solving these types of equations that are similar to methods used with polynomials. • Combining like terms • Taking square roots • Factoring Due to the periodic nature of the function, there are an infinite number of answers; so far we have just found the solutions over the interval [0, 2π).

5. Remember our rules… • Look for values on the unit circle • When taking the square root, don’t forget the + and – • Never divide by a function , move to the other side of the equation and factor!

6. Solving Equations – General Form sin x = ½, find all solutions Restricting the domain from [0, 2π) Look at the unit circle. x = {π/6, 5π/6} If we want all solutions with not just these solutions how can they be identified? We are going to use the period to help us…

7. sin x = ½ For the interval [0, 2π), x = {π/6, 5π/6} So the next two on the number line would be x = π/6 + 2π x = 5π/6 + 2π And the next two x = π/6 + 4π x = 5π/6 + 4π So we can write this in “general form” x = π/6 + 2nπ x = 5π/6 + 2nπ, where “n” is an integer. This gives us all solutions in both directions on the number line

8. Finding all solutions: sin x = ½

9. Solve, find all solutions 4sin2 x– 3 = 0 sin x – cos x sin x = 0 sin x (sec2 x + 1) = 0 2sin2 x= 2 + cos x

10. What about Tan and Cot? The period of tangent and cotangent is only π. To find the solutions for tangent and cotangent you can either • Find the solutions between [0, π) and then use the answer + nπ -or - • Find the solutions between [0, 2π) and then use the answer + 2nπbut you will have some redundant statements On a test you will see any multiple choice question written as option 1 above

11. Try these: tan2 x – 3 = 0 cos x (cot x + 1) = 0 (2sin2 x – 1) (tan x + 1) = 0

12. Solving equations involving multiple or partial angles Given: cos 2x = ½ “The angle when doubled has a cosine ratio value of ½.” First look for the solutions in the “reference” period [0, 2π) 2x = {π/3, 5π/3} What is the new period??? Hint: 2π/b

13. Solving equations involving multiple or partial angles Our new period is π, which also changes our interval between points An easy way to make the change to both is: 2x = {π/3, 5π/3} are the solutions Write the “general” form before changing the period2x = π/3 + 2nπ 2x = 5π/3 + 2nπ To find x, divide both sides by 2x = π/6 + nπx = 5π/6 + nπ This will adjust the period and the interval

14. Using your calculator

15. This method can be used for all functions Let’s try a couple more, one with multiple angles and one with a partial angle: 3tan2 2x – 1 = 0 2cos x/2 =1

16. Solving Equations for a Specific Domain cos2x = ½, find all [0, 2π) 2x = π/3 +2 nπx = π/6 + nπ 2x = 5π/3 +2 nπ x = 5π/6 +nπ Begin inserting the integers for “n” π/6 + (0)π = π/6 (in the interval requested) π/6 + (1)π = 7π/6 (in the interval requested) π/6 + (2)π =13π/6 (not in the interval requested) 5π/6 + (0)π = 5π/6 (in the interval requested) 5π/6 + (1)π = 11π/6 (in the interval requested) 5π/6 + (2)π =17π/6 (not in the interval requested) Solution set {π/6, 5π/6, 7π/6, 11π/6}

17. Solving Equations for a Specific Domain cos x/2 = ½, find all [0, 2π) x/2 = π/3 +2 nπx = 2π/3 + 4nπ x/2= 5π/3 +2 nπ x = 10π/3 +4nπ Begin inserting the integers for “n” 2π/3 + 4(0)π = 2π/3 (in the interval requested) 2π/3 + 4(1)π = 14π/3 (not in the interval requested) 10π/3 + (0)π = 10π/3 (not in the interval requested) Solution set {2π/3}

18. Try these: 4cos2 2x– 3 = 0 tan2 2x – 3 = 0 sin x/2 (cos x + 1) = 0

19. Try these For the solutions you found for the previous practice problems, find the solutions for the interval [0, 2π)

20. Notice We usually have 2 occurrences of a number in the unit circle (e.g., cos x = ½ in the first and fourth quadrant) cos 2x means we have two cycles in 2π (or the period for a cycle is π) Therefore we have twice as many answers in the 2π interval.

21. Homework 32 5.3 Page 376 # 19, 20, 33-38 all