# Thread: work out missing angle?

1. would you do inverse sin0.47= 28.03??

2. Why are you doing an inverse sin?

3. i wasn't really sure what to do. would this work:
cos/ 0.47= -2???

4. would it be cos/-0.47?
because you posted an earlier exapmle 4=6x, which is, 4/6

5. could i answer the question by doing this:
sin?/18 = sin/10
sin?=18xsin11/10=0.34 inverse sin= 20.09???

6. 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:

$\displaystyle f^{-1}(f(x))=f^{-1}(y).$

Here, all I've done is apply the inverse function of $\displaystyle f,$ denoted $\displaystyle f^{-1},$ to both sides. By the definition of function inverse, you now have

$\displaystyle f^{-1}(f(x))=x=f^{-1}(y).$

And there's your solution. Does that make sense? If so, how would you apply it to the equation

$\displaystyle \cos(x)=-0.47?$

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?

7. i don't understand that at all!

8. Ok, let's back up a minute. What is a function?

9. one quantity determins another quantity

10. 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?

11. i don't know

12. can i work it out like this:
just flip the cos over to the other side and make it inverse so it looks like this, ?=cos^-1(-0.47)??

13. 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.

Example:

$\displaystyle y=f(x)=x^{3}.$

The domain is all real numbers. The rule of association is the equation $\displaystyle f(x)=x^{3}.$

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

$\displaystyle f^{-1}(x)=\sqrt[3]{x}.$

In general, the way to find the inverse rule of association is to swap the symbols for $\displaystyle x$ and $\displaystyle y,$ and then solve for $\displaystyle y.$

Example: $\displaystyle y=f(x)=3x-4.$ The domain and the range are both the set of real numbers. Now I want to invert the function. First, swap $\displaystyle x$ and $\displaystyle y$ thus:

$\displaystyle x=3y-4.$ Now solve for $\displaystyle y:$

$\displaystyle x+4=3y$

$\displaystyle y=\dfrac{x+4}{3}=f^{-1}(x).$

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:

$\displaystyle f(x)=y$

$\displaystyle f^{-1}(f(x))=f^{-1}(y)$

$\displaystyle x=f^{-1}(y).$

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?

14. I don't really understand that. For the ? I got, ?=cos^-1(-0.47) =118.03

15. And you should double-check whether that answer is in degrees or radians, depending on which mode your calculator is set in.

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