1. Trigonometric Equations Reminders i) Radians Converting between degrees and radians:

2. ii) Exact Values 45 o right-angled triangle: 1 1 45 o Equilateral triangle: 1 2 2 1 60 o 30 o

3. degrees0o0o 30 o 45 o 60 o 90 o radians sin cos tan 0 0 1 0 1 0 1 Example: What is the exact value of sin 240 o ?

4. iii) Trigonometric Graphs: Amplitude Period 360 o 0 1 y x Period = 360 o Amplitude = 1

5. Period Amplitude 360 o 0 1 y x Period = 360 o Amplitude = 1

6. Period 0o0o 360 o 270 o 180 o 90 o y x Period = 180 o Amplitude cannot be defined.

7. Solving Trigonometric Equations Example: Step 1: Re-Arrange Step 2: consider what solutions are expected C A S T 0o0o 180 o 270 o 90 o All Sch…Talk Cr*!p

8. c a s t cos 3x is positive so solutions in the first and fourth quadrants x 3

9. Step 3: Solve the equation 3x = 60 o 300 o 420 o 660 o 780 o 1020 o 60 (360-60) (360+60) (720-60) (720+60) (1080-60) 1 st quadrant 4 th quadrant cos wave repeats every 360 o x = 20 o 100 o 130 o 220 o 260 o 340 o

10. Example: Step 1: Re-Arrange Step 2: consider what solutions are expected c a s t sin 6t is negative so solutions in the third and fourth quadrants x 6

11. Step 3: Solve the equation 6t = 225 o 315 o 585 o 675 o 945 o 1035 o 180+45 (360-45) (360+180+45) (720-45) (720+180+45) (1080-45) 3 rd quadrant 4 th quadrant sin wave repeats every 360 o t = 39.1 o 52.5 o 97.5 o 112.5 o 157.5 o 172.5 o

12. Example: The solution is to be in radians – but work in degrees and convert at the end. Step 1: Re-Arrange Step 2: consider what solutions are expected c a s t sin (2x – 60 o ) is positive so solutions in the first and second quadrants x 2

13. Step 3: Solve the equation 2x = 90 o or 210 o 2x-60 = 0 + 30 or (180 - 30)) 1 st quadrant 2 nd quadrant Now Add on the period of the wave to each of the values found in the first wave. i.e. x = 45 o + 180  or (105  + 180  ) X = 45  or 105 

14. Harder Example: Step 1: Re-Arrange Step 2: Consider what solutions are expected We need to solve 2 equations. 0o0o 360 o 270 o 180 o 90 o y x Expect 2 +ve solutions Expect 2 -ve solutions

15. Step 3: Solve the equation

16. Harder Example: Step 1: Re-Arrange Step 2: Consider what solutions are expected 360 o 0 1 y x We need to solve 2 equations. Just ONE solution Two solutions

17. Step 3: Solve the equations In the 1 st quadrant x = 19.5 o, 90 o, 160.5 o

18. Even Harder Example: Step 1: Re-Arrange Remember this ???? Step 2: Consider what solutions are expected We need to solve 2 equations. Just ONE solution Two solutions Step 3: Solve the equations c a s t