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)   (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)   (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 , ,   

f(x) = 9 - 18cos(x+)

9 - 18cos(x+) = 0

   -18cos(x+) = -9

      cos(x+) = 

The angles whose cosine is  have reference angle 60° in QI and QIV

x+  =  and x+ = 

where n is an integer.

Solve the first equation:

x+  = 

   x = 

We require that solutions be on the positive x-axis

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

Solutions for that are , , , 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+ = 

  x = 

  x = 

  x = 

  x = 

We require

 > 0

 > 0

4 + 6n > 0

    6n > -4

     n > 

     n > 

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

Solutions on the positive x-axis for are 

, , , etc.

or

, ,  , 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 .

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

The next one after that is the one that is approximately 10.5 or 

Answers: , , 

Edwin
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