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Trigonometric Equations

Trigonometric Equations. Reminders. i) Radians. Converting between degrees and radians:. 30 o. 2. 2. 1. 60 o. 45 o. 1. 1. 1. ii) Exact Values. 45 o right-angled triangle:. Equilateral triangle:. 0. 1. 0. 0. 1. 1. 0. Example:. What is the exact value of sin 240 o ?. y.

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Trigonometric Equations

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

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

  3. 0 1 0 0 1 1 0 Example: What is the exact value of sin 240o ?

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

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

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

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

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

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

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

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

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

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

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

  15. Step 3: Solve the equation

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

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

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

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