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

ii) Exact Values 45 o right-angled triangle: o Equilateral triangle: o 30 o

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

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

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

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

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

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

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) ( ) 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

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

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

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

Step 3: Solve the equation 2x = 90 o or 210 o 2x-60 = or ( )) 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  or (105   ) X = 45  or 105 

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

Step 3: Solve the equation

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

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

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