Why are you doing an inverse sin?
No, no, no. Cosine is a function. The notation cos(x) most emphatically does NOT mean cosine times x. It means "cosine of x". You are applying the cosine function to the number x.
Here's the more general case: suppose you have the equation f(x) = y, and you want to solve for x. How would you go about it? Well, what you would want to do, if you were able, would be to invert function f thus:
Here, all I've done is apply the inverse function of denoted to both sides. By the definition of function inverse, you now have
And there's your solution. Does that make sense? If so, how would you apply it to the equation
Let's use the variable x instead of ? from here on out. Less confusing. So you want to solve for x now. How would you do that?
I suppose that's a perfectly acceptable "rough-and-ready" definition. You should probably add in there something like this: "one quantity uniquely determines another quantity".
There are three aspects to a function: the domain, the range, and the rule of association. What are those, exactly?
Ok. Here's the terminology as I learned it.
A function is like a machine that converts objects of one kind into objects of another kind. The domain is all the allowed inputs to the function. The rule of association is the rule that tells you how the function converts the inputs into the outputs. The range is all the outputs that come, via the rule of association, from objects in the domain.
The domain is all real numbers. The rule of association is the equation
The range is all real numbers.
So the function inverse, if it exists (not all functions have inverses), gets you back to where you started. The domains and the ranges are flipped, the rule of association is inverted, etc. For the example above, the domain of the inverse is the range of the original function, and vice versa. The rule of association is
In general, the way to find the inverse rule of association is to swap the symbols for and and then solve for
Example: The domain and the range are both the set of real numbers. Now I want to invert the function. First, swap and thus:
Now solve for
Does that make sense?
The whole goal of this idea is to get back to where you started. That is, you want to be able to do this:
Thus, you can solve for various variables.
In your latest post, I think you have got it. You use the inverse cosine function, also known as the arccosine function, to solve for your question mark. What do you get?