SOLUTION: Find the first three x-intercepts of the graph of the given function on the positive x-axis. f(x) = 9 - 18 cos(x+π/3)

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Question 809218: Find the first three x-intercepts of the graph of the given function on the positive x-axis.
f(x) = 9 - 18 cos(x+π/3)


Found 2 solutions by lwsshak3, Edwin McCravy:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Find the first three x-intercepts of the graph of the given function on the positive x-axis.
***
f(x) = 9 − 18 cos(x+π/3)
set y=0
18cos(x+π/3)=9
cos(x+π/3)=9/18=-1/9
inverse cos(1/9)≈1.459455
..
1st intercept
x+π/3=1.459455
x=1.459455-π/3
x≈0.412258
..
2nd intercept
1.459455+2π=7.742640
x=7.742640-π/3
x=6.695442
..
3rd intercept
1.459455+4π=14.125826
x=14.125826-π/3
x=12.978628



Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor gave decimals and also missed solutions.  I think your teacher wanted
exact values, which as you will see are 4pi%2F3, 2pi, 10pi%2F3  

f(x) = 9 - 18cos(x+pi%2F3)

9 - 18cos(x+pi%2F3) = 0

   -18cos(x+pi%2F3) = -9

      cos(x+pi%2F3) = 1%2F2

The angles whose cosine is 1%2F2 have reference angle 60° in QI and QIV

x+pi%2F3  = pi%2F3+%2B+2n%2Api and x+pi%2F3 = 5pi%2F3+%2B+2n%2Api

where n is an integer.

Solve the first equation:

x+pi%2F3  = pi%2F3+%2B+2n%2Api

   x = 2n%2Api

We require that solutions be on the positive x-axis

2n*pi > 0, so n = 1,2,...

Solutions for that are 2pi, 4pi, 6pi, etc.

Those are approximately 6.3, 12.6, 18.8, etc.

(I'm using decimals only to tell which are the smallest three exact
solutions, not for final answers)
   
Solve the second equation:

x+pi%2F3 = 5pi%2F3+%2B+2n%2Api

  x = 4pi%2F3+%2B+2n%2Api

  x = 4pi%2F3+%2B+6n%2Api%2F3

  x = %284pi%2B6n%2Api%29%2F3

  x = expr%28%284%2B6n%29%2F3%29pi

We require

expr%28%284%2B6n%29%2F3%29pi > 0

expr%28%284%2B6n%29%2F3%29 > 0

4 + 6n > 0

    6n > -4

     n > -4%2F6

     n > -2%2F3

So n = 0, 1, 2, ...

Solutions on the positive x-axis for are 

expr%28%284%2B6%2A0%29%2F3%29pi, expr%28%284%2B6%2A1%29%2F3%29pi, expr%28%284%2B6%2A2%29%2F3%29pi, etc.

or

4pi%2F3, 10pi%2F3,  16pi%2F3, etc.

Those are approximately 4.2, 10.5, 16.8, etc.

(I'm using decimals only to tell which are the smallest three exact
solutions)

So the smallest is the one that is approximately 4.2, which is 4pi%2F3.

The next one after that is the one that is approximately 6.3 or 2pi.

The next one after that is the one that is approximately 10.5 or 10pi%2F3

Answers: 4pi%2F3, 2pi, 10pi%2F3

Edwin