Help with Trig Functions - min/max values and least + values of x

[TalkOrigin]

Hi, So I'm stuck on a part of trig which I can't seem to wrap my head around. I'm self teaching so no teacher to ask unfortunately. The question(s) come in the form:

"Find the the max and min value of each of the following functions. In each case, give the least positive values of x at which they occur. "

Then, a function would be given. For example:

e) $9 + sin(4x - 20)$

Another question (which I have not yet attempted) is:

f) $\frac{30}{2+cosx}$

I thought I was understanding how to find min and max values of equations. essentially, cosx/sinx will always be between -1 and 1, so you just put both those values in for x and figure it out. That worked a few times, but when I got to $9 + sin(4x - 20)$ it didn't work at all. I also am still not sure exactly what that equation is telling me, so I feel like even when I get the question right, I'm not truly understanding why. And I have no idea whatsoever about how to find least positive value for x without graphing it.

My attempt at a solution was to do

$9 + sin(4x1 - 20)$
$9 + -16$
$9 - 16 = = -7$

And I then used the same method except with -1, to get -15.

The problem is, the book I have doesn't tell you how to solve these functions, and just gives the answers without showing how to get there.

It also doesn't tell you what each number represents, and what to do when things are in parenthesis. I know, more or less, that the 9 at the beginning means that graph moves up the y-axis 9, or at least i thought I did until I put the function in on wolframalpha, and saw that it moves up 8. I also know that the number before cos/sin tells you the amplitude, and the x will tell you the period. I don't know how to approach the problem with the -20 and with things in parenthesis.

If I'm asking too much or there is too much to cover, then a suggestion of a specific book to buy/video to watch to clear this up would be great. I use Khan Academy videos, but for trig they are not very well structured in that they cover some advanced stuff before going back over basic stuff. Also I'm from the UK so the structure of what is taught and when is very different from the US I believe.

Anyway, thanks for any help, sorry for the long-winded post.

 

[CAF123]

If this is your first time learning this material, then I would recommend purchasing a textbook since Khan Academy in my opinion is not exhaustive.

For your first question, to find max/min of functions, usually you would just compute the derivative, set to zero and solve for x. However, if you have not done differentiation before, this can still be solved. As you said, the maximum value of sin is 1, so your function can never exceed the value 10. What value of x will make sin(4x -20) = 1?

Similarly for the other question.

 

[iRaid]

I honestly don't know how to do this problem without calculus..
Also, I don't understand what you did.

If you can use a calculators, you could graph it and find it. The answers to trig problems usually involve pi.

 

[tiny-tim]

Hi TalkOrigin!

e) $9 + sin(4x - 20)$

Another question (which I have not yet attempted) is:

f) $\frac{30}{2+cosx}$
…
My attempt at a solution was to do

$9 + sin(4x1 - 20)$
$9 + -16$
$9 - 16 = = -7$

And I then used the same method except with -1, to get -15.

No, you're doing 9 + sin(4x) - 20,

which you're saying lies between -11 - 4 and -11 + 4 = -15 and -7.

That's wrong for two reasons …

i] the "- 20" stays inside the sin function

ii] sin(4x) does not vary between ±4, it varies between ±1.

sin of any bracket lies between ±1 ​

how are you on the other one, 30/(2 + cosx)?

 

[TalkOrigin]

Wow, thanks for all the help already, this site is amazing.

So, I understand what you are all saying, that obviously whatever is in the brackets must be between -1 and 1. So I simply have to figure out what value for x will make $(4x - 20) = 1$ and $(4x - 20) = -1$ I can figure this out, but again, I'm not sure why. The values end up being $x = 5.25$ to get $(4x - 1) = 1$ , and $x=4.75$ for $(4x - 20) = -1$ . I'm with you loud and clear on that. However, how do I then use this info to find out least positive value of x at which they occur? The answer given is 27 1/2 degrees, and 72 1/2 degrees. This is actually a chapter after differentiation but I was advised by quite a few different people to get the chapter on trig done before I move on to calculus.

Hi tiny-tim :)

On the second question, $\frac{30}{2cosx}$ I think I'm fine with the min/max values of the function. As $2 + cosx$ is either $2 + 1$ or $2 + -1$, then it's just 30 divided by 3 and then 1, giving a max value of 30 and a min value of 10. Again, however, I'm not sure how I can know what the least positive value of x is for both of these.

Thanks again for all the help

 

[CAF123]

Wow, thanks for all the help already, this site is amazing.

So, I understand what you are all saying, that obviously whatever is in the brackets must be between -1 and 1. So I simply have to figure out what value for x will make $(4x - 20) = 1$

You want sin(4x - 20) = 1

Again, however, I'm not sure how I can know what the least positive value of x is for both of these.

Thanks again for all the help

What values of x makes cosx = 1? What values of x make cos x = -1 ?Then choose the smallest positive value in each case.

 

[TalkOrigin]

You want sin(4x - 20) = 1


What values of x makes cosx = 1? What values of x make cos x = -1 ?Then choose the smallest positive value in each case.

Ahhh, so you mean sin90 = 1, so I need to make (4x - 20) = 90. Meaning x = 27.5 right? Ahhh I finally think I understand :) Sort of haha.

 

[TalkOrigin]

Just wanted to say thank you to everybody, once I realized that I needed to find what value needs to be in the brackets so that sin ( ? ) = 1 or -1, it all clicked. Just did all the questions and got them all right, and they were driving me crazy earlier today haha.

Is there any way to close a thread so people know the issue has been resolved?

Thanks again