1. Sect. 5-3

2. What is SOLVING a trig equation? It means finding the values of x that will make the equation true. (Just as we did with algebraic equations!) Until now, we have worked with identities, equations that are true for ALL values of x. Now we’ll be solving equations that are true only for specific values of x.

3. Solving Trigonometric Equations Quadrant I Quadrant II Quadrant IIIQuadrant IV Cosine All Sine Tangent          Exact Values of Special Angles

4. Solving Trigonometric Equations 1)2) Solve for  if 0 ≤  < 2 . General Solutions Reference Angle Reference Angle General Solutions

5. 3) Solving Trigonometric Equations Solve for  if 0 ≤  < 2 . Reference Angle 4) Q1QII QIII QIV

6. 5) Solving Trigonometric Equations Solve for  if 0 ≤  < 2 . 6)

7. 1. Try to get equations in terms of one trig function by using identities. 6. Try to get trig functions of the same angle. If one term is cos2  and another is cos  for example, use the double angle formula to express first term in terms of just  instead of 2  3. Get one side equal to zero and factor out any common trig functions or reverse FOIL. 4. See if equation is in quadratic form and reverse FOIL. (Replace the trig function with x to see how it factors if that helps.) 5. If the angle you are solving for is a multiple of , don't forget to add 2  to your answer for each multiple of  since  will still be less than 2  when solved for. HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS 2. Be on the look-out for ways to substitute using identities

8. 7) 8) NO solution for cos  = 3. Reference Angle Solving Trigonometric Equations

9. 9) Solve 6 sin 2 x + sin x – 1 = 0; 0 º ≤ x < 360 º A quadratic equation! It may help to abbreviate sin x with s: i.e. 6s 2 + s – 1 = 0 Factoring this:(3s – 1)(2s + 1)= 0    α = 19.47 º α = 30 º So, x = (To nearest 0.1 º.) 19.5 °, 160.5 °, 210 °,330 °.

10. You Try… Solve for y 10)

11. Solve for y: domain We have to give all the answers

12. 11) The equation cannot be factored. Therefore, use the quadratic equation to find the roots: Reference Angles: Solving Trigonometric Equations

13. Using a Graphing Calculator to Solve Trigonometric Equations Therefore,  and 

14. Use the Pythagorean Identity to replace this with an equivalent expression using sine. Combine like terms, multiply by -1 and put in descending order Factor (think of sin  like x and this is quadratic) Set each factor = 0 and solve 12)

15. Solve for x where the domain is 13)

16. Use a graphing utility to solve the equation. Express any solutions rounded to two decimal places. Graph this side as y 1 in your calculator Graph this side as y 2 in your calculator You want to know where they are equal. That would be where their graphs intersect. You can use the trace feature or the intersect feature to find this (or these) points (there could be more than one point of intersection). There are some equations that can't be solved by hand and we must use a some kind of technology. 14)

17. This was graphed on the computer with graphcalc, a free graphing utility you can download at www.graphcalc.com After seeing the initial graph, lets change the window to get a better view of the intersection point and then we'll do a trace. Rounded to 2 decimal places, the point of intersection is x = 0.53 check: This is off a little due to the fact we approximated. If you carried it to more decimal places you'd have more accuracy.

18. YOU TRY… Solve Note: There is no solution here because  2 lies outside the range for cosine. 2)

19. Closure Discuss the similarities and differences in the steps for solving a trigonometric equation versus solving a polynomial equation.

20. Solving Trigonometric Equations with Multiple Angles Solve: Solution: Since 3x refers to an angle, find the angles whose cosine value is ½. Now divide by 3 because it is angle equaling angle. Notice the solutions do not exceed 2 . Therefore, more solutions may exist. Return to the step where you have 3x equaling the two angles and find coterminal angles for those two. Divide those two new angles by 3. 1)

21. Solving Trigonometric Equations The solutions still do not exceed 2 . Return to 3x and find two more coterminal angles. Divide those two new angles by 3. The solutions still do not exceed 2 . Return to 3x and find two more coterminal angles. Divide those two new angles by 3. Notice that 19  /9 now exceeds 2  and is not part of the solution. Therefore the solution to cos 3x = ½ is

22. 2) Solve the equations.

23. Example 3) Solve

24. x y Take the fourth root of both sides to obtain: cos(2x)= ± From the unit circle, the solutions for 2  are 2  = ± + kπ, k any integer. π 6 Example 4: Find all solutions of cos 4 (2x) =. 9 16 Answer:  = ± + k ( ), for k any integer. 12 π 2 π 1 π 6 -π-π 6 x = - x = π π

25. 5) Solve

26. Example 6:Solve 3 + 5 tan 2x = 0; 0 º ≤ x ≤ 360 º. Firstly we need to make tan 2x the subject of the equation: The tangent of an angle is negative in the = 30.96 ° The range must be adjusted for the angle 2x.i.e. 0 ° ≤ 2x ≤ 720 °. Hence: 2x = x = 74.5 °, 164.5 °, 254.5 °, 344.5 °.   149.04 °, 329.04 °, 509.04 °,689.04 °. 2 nd and 4 th quadrants. tan 2x = – 3535

27. Example 7:Solve 2 sin (4x + 90 º ) – 1 = 0; 0 < x < 90 º The sine of an angle is positive in the The range must be adjusted for the angle 4x + 90 º. i.e. 0 º < 4x < 360 º 90 º < 4x + 90 º < 450 º 4x + 90 º = 4x = 60 º, 300 º x = 15 º, 75 º   = 30 º 150 °,390 ° 1 st and 2 nd quadrants. sin (4x + 90 º )

28. Solving Trigonometric Equations Try these: 1. 2. 3. 4. Solution

29. Sect. 5-3 #’s 61, 63, 65,

30. Evaluate. a) b) Finding Exact Values

31. Solving Trigonometric Equations Solve: X Why is 3  /2 removed as a solution? 26)

32. 5.5 Trigonometric Equations Objectives –Find all solutions of a trig equation –Solve equations with multiple angles –Solve trig equations quadratic in form –Use factoring to separate different functions in trig equations –Use identities to solve trig equations –Use a calculator to solve trig equations

33. Is this different that solving algebraic equations? Not really, but sometimes we utilize trig identities to facilitate solving the equation. Steps are similar: Get function in terms of one trig function, isolate that function, then determine what values of x would have that specific value of the trig function. You may also have to factor, simplify, etc, just as if it were an algebraic equation.

34. 5) QIQIV 8) Reference Angle

35. 14) Reference Angle: Therefore: