composite trig

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Is there a way to algebraically find compositions such as

or ?

I can't find any identities relating tan to either sin or cos, which would allow it to do something like this:



Also I don't understand why and not .

I would have thought that since we can define , it would be more sensible to have

OR even better:

Thanks

(Oh also one more thing, when we sketch and and is it implied that the domain and range are:
; ; and respectively?
Why doesn't have range ? )

[Mao]

I shall answer questions in no particular order =]


1. why does

This is because has a range of (it is defined to that).

2. why are the domains and ranges of inverse circular functions what they are?

the circular functions are periodic, hence they repeat themselves every so often. The problem with this is that it won't be one-to-one if we didn't restrict the domain. Sketch the three circular functions to see for yourself. When you deal with the inverse circular functions, remember that they are ONLY what they are defined to be, nothing more.

3. how to do and things similar to this?
construct a triangle, first concerning tan(x) [the opposite is x, adjacent is 1, using pytha to find the hypotenuse to be
now, we take the inverse tan of x, i.e. we get the angle between adj and hyp, taking the cos of that, we use

4. how to do ?

firstly, unless both and are tan, cannot be tan. This is because [and you have learnt composite functions] the range of inner must be a subset of the domain of the outer, which tan is not, R does not fit in [-1,1].
and your method dealing with this when they are sin/cos are quite correct.

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Ah, thanks Mao! Now i think i understand

[Mao]

Another more algebraic method for number 3, use the pythag identity :

e.g.

e.g.





(negative solution discarded since inverse tan's values lies in first and fourth quadrant.

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Cool! Substituting is quite inspired   

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So, leading on from the range of being defined as ,

If you're asked to find the inverse of ,

Would technically be incorrect? I guess I am messing with predefined ranges after all...

The answers gave it as:

where

But it seems all drawn-out and complicated - is it all necessary?

Thanks again

[humph]

So, leading on from the range of being defined as ,

If you're asked to find the inverse of ,

Would technically be incorrect? I guess I am messing with predefined ranges after all...

The answers gave it as:

where

But it seems all drawn-out and complicated - is it all necessary?

Thanks again

It'd be better to say where , than say , as is already a function defined to have a different range. But other than that it's fine - you can be specific about the range, unlike the answers.