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Mastering Trigonometric Equations and Graphs

Understand trigonometric equations, convert between degrees and radians, solve graphs, and find exact values in triangles. Learn how to solve equations in different quadrants and enhance your trigonometry skills.

wilhelmina-carter
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Mastering Trigonometric Equations and Graphs

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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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