How do I do this calculation involving the SIN function?

[Francis Bacon]

TL;DR Summary

Time t value of SIN function should be calculated to get a SIN function value of -1

Hi,
I am new here and hope I have posted my thread in the right forum.

I have the following SIN function in Excel: =1*(SIN(2*PI()*1,6667*0,45))

The result is -1. That is what I want, so no problem.

But what I want is a function that calculates the Time t value, in this example the value 0.45, that results the SIN value -1.

So the function is: 1*(SIN(2*PI()*1,6667* x )) = -1.

How can I calculate x (Time t)?

Thank you all very much for your assistance.

 

[BvU]

Hello @Francis Bacon ,

!
​

So what you want is the inverse of the sine function. For -1 you get ##\arcsin (-1) = -{\pi\over 2} \ ##. The equation ##\sin\pi = -1\ ## has solutions modulo ##2\pi##, so $$\sin y = -1 \quad \Leftrightarrow \quad y = -{\pi\over 2} + 2n \pi \ $$in your case, with ##n = 1\ ## you get ##{10 \over 3}\pi x = {3\over 2} \pi \quad \Leftrightarrow \quad x = {9\over 20}##

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[FactChecker]

To calculate x, you need the inverse of the SIN function, the ASIN function, which returns a value between ##-\pi/2## and ##\pi/2##. Try $$x = ASIN(-1)/(2*PI()*1,6667)$$

Keep in mind that there are several values of x that will work. They differ by multiples of 1/1,6667 = 0.6. The value you get may not be the one you expected. When I do this calculation, I get x= -.15 and -1.5+0.6= your 0.45

 

[Francis Bacon]

Hi "BvU" and "FactChecker",

really thank you very much for your quick and extremely helpful answers. I am blown away. Thanks again.

 

[Francis Bacon]

Hi,

In light of the very good answers I have received to my question, I would like to ask another question, but this time perhaps a more complicated one.

I have attached two files to this question to hopefully make it a little easier to answer my question.

I have created a sine function with a skew factor (see my Excel file).

The formula for the skew factor can be found in the PDF file.

What I want to achieve is a sine function that always starts at the value -1, no matter if the sine function has a skew factor or not.

As you can see from the Excel examples, the value for the variable "Time t" (in my Excel file, t is labeled "Dummy for Cycle Length") is always different, depending on how long the cycle is.

But as already answered above, the value for t can be calculated with the inverse sine function IF the sine function has NO skew factor.

If the sin function has a skew factor, things get quite complicated.

I have found a solution that "works" for a lay mathematician.
The formula is: =ASIN(-1)/(2*PI()*1.1111) - (((90/60)*0.5)/10)

Cycle Length: 90
Frequency f = 1.1111 (1/90 * 100)
Skew Factor: 0.5
Value 60 is a constant

You can find this formula in the cell after the cell called "Dummy Start at Cycle Trough with Skew:".

The special part of the formula is of course the second half: "- (((90/60)*0.5)/10)".

This is my layman mathematician solution.

Now my question: "Is it possible to make this second half of the formula more professional?"

The problem with my layman mathematician solution is that when the skew factor is 0.58 or greater, the start value of the sine function is no longer exactly -1. With a skew factor of 0.57 or less, my lay mathematician solution "works".

I would like to mention that it is important, that both the sine function with a skew factor and the sine function without a skew factor, must make their respective cycle trough at the same time.

There must be no shift of the cycle trough between the sine function with a skew factor and the sine function without a skew factor.

I hope my description is understandable and thank you very much for your efforts.

 

[BvU]

Hello again,

If I try to translate the job you have in mind for us, then:

You want to know when ##\ \operatorname{SineSkewed}(x, s_\text{kew}) = -1 ##, right ?
So you want to solve for ##x## $$\operatorname{SineSkewed}(x, s_\text{kew}) =
{\sin x\over \sqrt{(s_\text{kew} + \cos x)^2 + (\sin x)^2 }}= -1 $$and the conditions for that to happen are ##\ \sin x < 0\ ## .AND. ##\ s_\text{kew} + \cos x = 0\ ##.
In the range ##\ [0, 2\pi]\ ## the first comes out at ##\ \pi < x < 2\pi \ ## and the second is when ## x = - \arccos(-s_\text{kew})\ ##.

In your case the 'professional' expression for Dummy start at cycle with skew ($N$2) is then
##\quad## =ACOS(-U2)/2/PI()/L2

For a skew factor of 0.57 I get 0.3119 in $N$2, very close to what you had.
And it works for ## s_\text{kew} = -0.999999999999 ## to ## s_\text{kew} = 1 ## (so not for -1 ! -- can you see why not ? )

Job completed !

##\ ##

 

[Francis Bacon]

Hi BvU,

thanks for your work and especially for describing my issue in a clear mathematical way.

Unfortunately I have some problems with your formula: =ACOS(-U2)/2/PI()/L2

I have tried some variations of your formula but without success.

What do you mean with (-U2), L2, and about 2/PI() ?

Thanks again.

 

[BvU]

Oops, confusion on my end (had inserted a few columns). I meant that cell N2 contains
##\qquad## =ACOS(-Q2)/2/PI()/H2
Q2 is the skew factor, H2 is frequency * 100

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[Francis Bacon]

Hi BuV,

Thank you very much. Your formula works perfectly.

I don't know why I didn't figure out the problem with the column letters. Maybe I tried too many variations and in the end everything was messed up.

Let me add one last point. It is necessary to multiply the results by -1 to get a sine function that starts with -1.

So your modified formula therefore is: =(ACOS(-Q2)/2/PI()/H2)*-1

Thank you. Math is really a great thing.

 

[BvU]

multiply the result by -1

Correct. I had that in B3 (=##\;-##N2) and overlooked that as well

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