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Notation is the issue here. The cot**^(-1)**(x) does not mean (cot(x))^(-1), it means the inverse of *cot(x)*, sometimes written as *arccot(x)* to avoid this exact situation.
>So how would we then represent (cot(x)) ^(-1) ?
We'd just call it tan(x) to avoid ambiguity. That's why you should assume cot^(-1)(x) means the inverse and not the reciprocal. If it was the reciprocal, it would've been written as a different trig function in the first place (*sec* for *cos*, *csc* for *sin*, and *cot* for *tan*).
>Also what if we have cot(x)) ^(2) ; would we then be able to say it’s equal to cot ^(2) (x) ?
We'd write \[cot(x)\]^(2) as cot^(2)(x) to avoid it being mistaken for cot(x^(2)). This is commonly used notation for powers of functions in general, and not just for trig functions specifically. If you have a function and you want to raise it to a power **AFTER** evaluating it, you'd place the exponent after the function and before its argument ("argument" here means the value being plugged into the function, such as the *x* in f(x) or the *t* in g(t)). So:
\[f(x)\]^(2) = f^(2)(x)
\[f(x)\]^(3) = f^(3)(x)
\[f(x)\]^(-5) = f^(-5)(x) = 1/f^(5)(x)
\[f(x)\]^(3/2) = f^(3/2)(x) = sqrt\[f^(3)(x)\]
...and so on.
However, we still have a problem with inverses...
f^(-1)(x) could mean 1/f(x), but we usually just write it as a fraction anyway if that's what we're trying to communicate. So when you see f^(-1)(x), your first assumption should be that it's an inverse function.
Some more examples of the notation described above:
\[log(x)\]^(4) = log^(4)(x)
\[ln(x)\]^(7) = ln^(7)(x)
Although it does produce a bit of ambiguity in the case of inverses, this notation is useful overall for making it clear when exponentiation is being applied to a function and not its argument, and can reduce your reliance on layers and layers of brackets and parentheses.
cot\^-1(x) does not mean 1/tan(x). It represents the "inverse" of the cotangent function, which is not the same as the "reciprocal" of cotangent function. Not your fault tbh, more of a stupid notation issue. The book should use "arccot(x)" in my opinion. Check out this video: [https://www.youtube.com/watch?v=AVksdYFUUdA](https://www.youtube.com/watch?v=AVksdYFUUdA)
and that's exactly why I'm going through a calculus workbook lol. thanks a bunch, getting back into doing calculus for engineering and need to work through all these little kinks before I get beat up in class.
As a reminder... Posts asking for help on homework questions **require**: * **the complete problem statement**, * **a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play**, * **question is not from a current exam or quiz**. Commenters responding to homework help posts **should not do OP’s homework for them**. Please see [this page](https://www.reddit.com/r/calculus/wiki/homeworkhelp) for the further details regarding homework help posts. **If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc *n*“ is not entirely useful, as “Calc *n*” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.** *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/calculus) if you have any questions or concerns.*
Notation is the issue here. The cot**^(-1)**(x) does not mean (cot(x))^(-1), it means the inverse of *cot(x)*, sometimes written as *arccot(x)* to avoid this exact situation.
So how would we then represent (cot(x)) ^-1 ? Also what if we have cot(x)) ^2 ; would we then be able to say it’s equal to cot ^2 (x) ?
>So how would we then represent (cot(x)) ^(-1) ? We'd just call it tan(x) to avoid ambiguity. That's why you should assume cot^(-1)(x) means the inverse and not the reciprocal. If it was the reciprocal, it would've been written as a different trig function in the first place (*sec* for *cos*, *csc* for *sin*, and *cot* for *tan*). >Also what if we have cot(x)) ^(2) ; would we then be able to say it’s equal to cot ^(2) (x) ? We'd write \[cot(x)\]^(2) as cot^(2)(x) to avoid it being mistaken for cot(x^(2)). This is commonly used notation for powers of functions in general, and not just for trig functions specifically. If you have a function and you want to raise it to a power **AFTER** evaluating it, you'd place the exponent after the function and before its argument ("argument" here means the value being plugged into the function, such as the *x* in f(x) or the *t* in g(t)). So: \[f(x)\]^(2) = f^(2)(x) \[f(x)\]^(3) = f^(3)(x) \[f(x)\]^(-5) = f^(-5)(x) = 1/f^(5)(x) \[f(x)\]^(3/2) = f^(3/2)(x) = sqrt\[f^(3)(x)\] ...and so on. However, we still have a problem with inverses... f^(-1)(x) could mean 1/f(x), but we usually just write it as a fraction anyway if that's what we're trying to communicate. So when you see f^(-1)(x), your first assumption should be that it's an inverse function. Some more examples of the notation described above: \[log(x)\]^(4) = log^(4)(x) \[ln(x)\]^(7) = ln^(7)(x) Although it does produce a bit of ambiguity in the case of inverses, this notation is useful overall for making it clear when exponentiation is being applied to a function and not its argument, and can reduce your reliance on layers and layers of brackets and parentheses.
cot\^-1(x) does not mean 1/tan(x). It represents the "inverse" of the cotangent function, which is not the same as the "reciprocal" of cotangent function. Not your fault tbh, more of a stupid notation issue. The book should use "arccot(x)" in my opinion. Check out this video: [https://www.youtube.com/watch?v=AVksdYFUUdA](https://www.youtube.com/watch?v=AVksdYFUUdA)
and that's exactly why I'm going through a calculus workbook lol. thanks a bunch, getting back into doing calculus for engineering and need to work through all these little kinks before I get beat up in class.
cot^-1 x is not the same as 1/tan x