Hi :)
I was just smashing out some maths study and it was going pretty well until I got to some worked examples that have confused me.
My textbook seems to contradict itself with two worked examples, so now I'm unsure of the correct procedure.
The questions are:
Solve the following equations for x
a) 2sin(2x) - 1 = 0, 0 ≤ x ≤ 2π
working out
Change domain
2sin(2x) - 1 = 0, 0 ≤ 2x ≤ 4π
2sin(2x) = 1
sin(2x) = 1/2
(USE SPECIAL TRIANGLE AND UNIT CIRCLE TO FIND EXACT VALUES)
^ the base ended up being π/6
2x = 0 + π/6, π - π/6, 2π + π/6, 3π - π/6
2x = π/6, 5π/6, 13π/6, 17π/6
Get rid of the 2
x = π/12, 5π/12, 13π/12, 17π/12
Note: at the end we got rid of the 2 by applying it to the denominator of the fraction
The next example was:
b) 2cos(2x - π) - 1 = 0, -π ≤ x ≤ π
working out
Change domain
2cos(2x - π) - 1 = 0, -3π ≤ 2x - π ≤ π
2cos(2x - π) = 1
cos(2x - π) = 1/2
(USE SPECIAL TRIANGLE AND UNIT CIRCLE TO FIND EXACT VALUES)
^ the base ended up being π/3
2x - π = -2π - π/3, -2π + π/3, 0 - π/3, 0 + π/3
2x - π = -7π/3, -5π/3, -π/3, π/3
2x = -7π/3 + π, -5π/3 + π, -π/3 + π, π/3 + π
2x = -4π/3, -2π/3, 2π/3, 4π/3
x = -2π/3, -π/3, π/3, 2π/3
Note: at the end we got rid of the 2 by applying it to the numerator of the fraction
So now I am confused as to whether we apply the changes to the denominator or numerator? My textbook changes its mind a couple times and will sometimes apply it to the denominator and other times the numerator.. Is it just a matter of which one is easiest to change? i.e. if you had 2x = 3π/4 you would make it 3π/8 instead of 1.5π/4 because it is "easier"??
Thanks! :)