SOLUTION: f(x)=2sin(1/2x+5pi/6) Find five exact value points on the graph

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Question 716705: f(x)=2sin(1/2x+5pi/6)
Find five exact value points on the graph
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since all multiples of pi%2F6 are special angles and since the constant part of the argument, 5pi%2F6, is already a multiple of pi%2F6, it will probably be easiest if we choose values for x that would make the other term in the argument, %281%2F2%29x, also be a multiple of pi%2F6. By doing so we would then be able to add the fractions and end up with an argument that is also a multiple of pi%2F6.

Perhaps you can already tell what values x should have that would turn %281%2F2%29x into a multiple of pi%2F6. If not, then just pick a multiple of pi%2F6, set %281%2F2%29x equal to it and solve for x. Let's be easy on ourselves and choose pi%2F6 itself:
%281%2F2%29x+=+pi%2F6
Multiplying both sides by 6 we get:
3x+=+pi
Dividing by 3 we get:
x+=+pi%2F3

Perhaps you can now see that if x is any multiple of pi%2F3, then %281%2F2%29x would end up being some multiple of pi%2F6. I'll get you started on a table of values:
  x    (1/2)x   (1/2)x+5pi/6  sin((1/2)x+5pi/6)   2sin((1/2)x+5pi/6)   f(x)
pi/3    pi/6      6pi/6 = pi        0                  0                0
-pi/3   -pi/6   4pi/6 = 2pi/3     sqrt(3)/2            sqrt(3)         sqrt(3)
  0       0         5pi/6          1/2                 1                1
I'll leave it up to you to find two more.

P.S. We did not have to make x be a value that made the argument of sin be a multiple of pi/6. We just did that because it made things easier. Any x that would make (1/2)x+5pi/6 turn out to be any of the special angles would work.