1. 41: Trig Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. Trig Equations Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. Trig Equations BUT, by considering the graphs of and, we can see that there are many more solutions: e.g.1 Solve the equation. Solution: The calculator gives us the solution x = Every point of intersection of and gives a solution ! In the interval shown there are 10 solutions, but in total there are an infinite number. The calculator value is called the principal solution principal solution

4. Trig Equations 1 We will adapt the question to: Solution: The first answer still comes from the calculator: Add the line Solve the equation for Sketch between There are 2 solutions. The symmetry of the graph...... shows the 2 nd solution is It’s important to show the scale. Tip: Check that the solution from the calculator looks reasonable.

5. Trig Equations 1 Solution: The first answer from the calculator is Add the line e.g. 2 Solve the equation in the interval Sketch between There are 2 solutions. The symmetry of the graph...... shows the 2 nd solution is

6. Trig EquationsSUMMARY Find the principal solution from a calculator. Find the 2 nd solution using symmetry where c is a constant  To solve orfor or Draw the line y = c. Sketch one complete cycle of the trig function. For example sketch from to.

7. Trig Equations 1 Exercises The 2 nd solution is 1.Solve the equations (a) and (b) for Solution: (a) ( from calculator )

8. Trig Equations 1 Solution: ( from calculator ) The 2 nd solution is (b) Exercises

9. Trig Equations 2 -2 More Examples e.g. 3 Solve the equation in the interval giving answers correct to the nearest whole degree. Solution: ( from the calculator ) The 2 nd solution is

10. Trig Equations 2 -2 So all solutions to the equation can be found by repeatedly adding or subtracting to the first value. Notice that the period of is and there is only one solution to the equation in each interval of.

11. Trig Equations So, to solve for Principal solution: This process is easy to remember, so to solve there is no need to draw a graph. First subtract Now add to and keep adding... Ans:

12. Trig Equations Solve the equation for Exercise Solution: Principal value Ans: Adding

13. Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 d. p. ( Because of the interval, it’s convenient to sketch from to. ) Switching the calculator to radians, we get Solution: implies radians 2 nd solution: Ans: More Examples

14. Trig EquationsSolution: ( from the calculator ) e.g. 5 Solve the equation for This value is outside the required interval...... but we still use it to solve the equation. Tip: Bracket a value if it is outside the interval. We extend the graph to the left to show More Examples

15. Trig Equations 1 e.g. 5 Solve the equation for Since the period of the graph is this solution...... is Solution: More Examples

16. Trig Equations 1 Solution: e.g. 5 Solve the equation for Symmetry gives the 2 nd value for. The values in the interval are and More Examples

17. Trig Equations So, if more solutions are required we add ( or subtract ) to those we already have. The graphs of and repeat every. For solutions in the interval, we also haveand e.g. In the previous example, we had )3600(   x

18. Trig Equations 1 Solution: Principal value e.g. 6 Solve for By symmetry, Method 1 Ans: Subtract from : ( is outside the interval )

19. Trig Equations 1 Solution: Principal value e.g. 6 Solve for The solution can be found by using the symmetry of about the y -axis Method 2 Ans: Add to :

20. Trig EquationsSUMMARY  To solve or Once 2 adjacent solutions have been found, add or subtract to find any others in the required interval. Find the principal value from the calculator. Sketch the graph of the trig function showing at least one complete cycle and including the principal value. Find a 2 nd solution using the graph.  To solve Find the principal value from the calculator. Add or subtract to find other solutions.

21. Trig Equations 1.Solve the equations ( giving answers correct to the nearest whole degree ) (b) for (a) for Exercises

22. Trig Equations 1 (a) for Solution: Principal value By symmetry, Ans: Exercises

23. Trig Equations 1 1 Ans: (b) for Solution: Principal value Either:Or: Exercises

24. Trig Equations

25. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

26. Trig Equations Solution: The first answer comes from the calculator: Add the line Solve the equation for Sketch between There are 2 solutions. The symmetry of the graph...... shows the 2 nd solution is e.g. 1

27. Trig Equations Solution: The first answer from the calculator is Add the line e.g. 2 Solve the equation in the interval Sketch between There are 2 solutions. The symmetry of the graph...... shows the 2 nd solution is

28. Trig Equations e.g. 3 Solve for Principal solution: This process is easy to remember, so to solve there is no need to draw a graph. First subtract Now add to and keep adding... Ans:

29. Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 d. p. ( Because of the interval, it’s convenient to sketch from to. ) Switching the calculator to radians, we get Solution: radians 2 nd solution: Ans:

30. Trig Equations Solution: ( from the calculator ) e.g. 5 Solve the equation for This value is outside the required interval...... but we still use it to solve the equation. Tip: Bracket a value if it is outside the interval. We extend the graph to the left to show

31. Trig Equations Symmetry gives the 2 nd value as Ans:, Since the period of the graph is, the 1 st solution in is

32. Trig Equations Solution: Principal value e.g. 6 Solve for By symmetry, Method 1 Ans: Subtract from : ( is outside the interval )

33. Trig Equations The solution can be found by using the symmetry of about the y -axis Method 2 Ans: Add to :

34. Trig Equations SUMMARY  To solve or Once 2 adjacent solutions have been found, add or subtract to find any others in the required interval. Find the principal value from the calculator. Sketch the graph of the trig function showing at least one complete cycle and including the principal value. Find a 2 nd solution using the graph.  To solve Find the principal value from the calculator. Add or subtract to find other solutions.