Trigonometric Equations Reminders i Radians Converting between degrees

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

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

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

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

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

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

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

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

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

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

Step 3: Solve the equation 6 t = 225 o 315 o 585 o 675 o 945 o 180+45 (360 -45) (360+180+45) (720 -45) (720+180+45) 3 rd quadrant t = 39. 1 o 4 th quadrant 52. 5 o 97. 5 o 1035 o (1080 -45) sin wave repeats every 360 o 112. 5 o 157. 5 o 172. 5 o

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 s a t c sin (2 x – 60 o ) is positive so solutions in the first and second quadrants x 2

Step 3: Solve the equation 2 x-60 = 0 + 30 or (180 - 30)) 2 x = 90 o or 210 o X = 45 or 105 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 )

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

Step 3: Solve the equation

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

Step 3: Solve the equations In the 1 st quadrant x = 19. 5 o , 90 o , 160. 5 o

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