Finding specific values for the equation "y=9cos (2π/3 (x - 2)) +16"

[NZG]

In this question, I was asked to find the first two (positive) x-values of the equation "y=9cos (2π/3 (x - 2)) +16", where y was equal to 20. So, obviously, I plugged in 20 as y. I got one value: 2.52, which I'm fairly sure is correct. However, this is the second positive value, as I know from looking at the graph (thank you, Desmos!) The first positive value (again, which I know from looking at the graph) is a little under 1.5, probably 1.47 or 1.48. HOW DO I FIND THIS?

Neither are mid/high/low points- they are two points on opposite sides of the same curve so the distance between them is NOT the period, but around 1.02. I've tried similar problems and I can't seem to get that elusive first value- what am I doing wrong?

Any help would be GREATLY appreciated.

 

[Steven G]

y=9cos (2π/3 (x - 2)) +16
Since y=20, we get 20=9cos (2π/3 (x - 2))+16
Now since 20 = 4 + 16 we have 9cos (2π/3 (x - 2)) =4.
Since 9*4/9 =4, we arrive at cos (2π/3 (x - 2)) = 4/9
So one solution come from arccos(4/9) = 2π/3 (x - 2)
Then 3/(2π)arccos(4/9) = (x - 2)
Yielding x= 3/(2π)arccos(4/9) +2 as one solution
Can you find another answer? If not, then come by, tell us why and then you get a hint.

 

[Dr.Peterson]

In this question, I was asked to find the first two (positive) x-values of the equation "y=9cos (2π/3 (x - 2)) +16", where y was equal to 20. So, obviously, I plugged in 20 as y. I got one value: 2.52, which I'm fairly sure is correct. However, this is the second positive value, as I know from looking at the graph (thank you, Desmos!) The first positive value (again, which I know from looking at the graph) is a little under 1.5, probably 1.47 or 1.48. HOW DO I FIND THIS?

Neither are mid/high/low points- they are two points on opposite sides of the same curve so the distance between them is NOT the period, but around 1.02. I've tried similar problems and I can't seem to get that elusive first value- what am I doing wrong?

Any help would be GREATLY appreciated.

It will help if you show your actual work, so we can point out what you could have done differently.

I suspect that you found the first value greater than 2 (related to the fact that the function has been shifted right by 2 units, as indicated by "(x-2)").

To find the first positive value, you need to look to the left of 2; that is (assuming your work was generally similar to Steven G's), you need a negative angle whose cosine is 4/9. What would that be?

(One way to approach the whole problem is to find all solutions, and use that result to find the one you want. Do you know how to do that?)

 

[The Highlander]

In this question, I was asked to find the first two (positive) x-values of the equation "y=9cos (2π/3 (x - 2)) +16", where y was equal to 20. So, obviously, I plugged in 20 as y. I got one value: 2.52, which I'm fairly sure is correct. However, this is the second positive value, as I know from looking at the graph (thank you, Desmos!) The first positive value (again, which I know from looking at the graph) is a little under 1.5, probably 1.47 or 1.48. HOW DO I FIND THIS?

Neither are mid/high/low points- they are two points on opposite sides of the same curve so the distance between them is NOT the period, but around 1.02. I've tried similar problems and I can't seem to get that elusive first value- what am I doing wrong?

Any help would be GREATLY appreciated.

There's not much point in being "fairly sure" something is "correct" in Maths; either it is or it isn't.
Even the graphing App you're so fond of gives the "first result" you mention as 2.53 not 2.52 which is what you do get (to 2 decimal places) if you carry out the required calculation (as demonstrated by @Steven G, above).
You need to be a bit more thorough in your approach. Playing with graphing tools is all very well and may offer valuable insights but you need to carry out the relevant calculations (accurately) to achieve the stated objective. The two previous posts provide everything you ought to need to do so.

 

[Deleted member 4993]

In this question, I was asked to find the first two (positive) x-values of the equation "y=9cos (2π/3 (x - 2)) +16", where y was equal to 20. So, obviously, I plugged in 20 as y. I got one value: 2.52, which I'm fairly sure is correct. However, this is the second positive value, as I know from looking at the graph (thank you, Desmos!) The first positive value (again, which I know from looking at the graph) is a little under 1.5, probably 1.47 or 1.48. HOW DO I FIND THIS?

Neither are mid/high/low points- they are two points on opposite sides of the same curve so the distance between them is NOT the period, but around 1.02. I've tried similar problems and I can't seem to get that elusive first value- what am I doing wrong?

Any help would be GREATLY appreciated.

I got one value: 2.52

You say one value: 2.52 - one value of what? You got value of 'y' or 'x'. I can make a guess - but you should be clear.

What was the dimension of '2.52'?

You have not posted the ORIGINAL problem. In the original problem - Was the domain specified?