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I suspect that you are confusing things a bit.
The inverse sine, called the arcsine is a function defined in the closed interval [-1,1]. And so is the inverse cosine. Any value outside this interval will give you a non-real result (meaning a complex one).
There are no limitations on the domain of definition of the inverse hyperbolic sine or sinh^-1
If your input value is allowed to be complex, the arcsine function gives a complex value. See the screen capture
Does it refuse to do so or does it give an error message? Three common errors:
Not having the correct angle unit. Wrong result, No error message
Confusing reciprocal of sine (1/sin(x) with arc sine (x) ,sin^-1(x). Confusing the reciprocal of cosine, 1/cos(x) with arc cosine (cos^-1(x)). Wrong result, No error message
Taking the argument of the inverse sine and/or inverse cosine functions outside the interval [-1,1]. This gives a domain error.
The sine and cosine function have a range between [-1, 1]. The domain of their inverse functions is [-1,1]. So 20/1 which is 20 is out of the domain of definition of the functions. No limitations for tangent and cotangent.
The range of the cosine and sine functions is limited to the interval [-1, 1]. As a result the calculator will send an error message if the functions are called with arguments outside the interval [-1, 1]. Aside from this, I do not see any reason why the functions would give an error.
The inverse sine and cosine equations mean "angle who's sine/cosine is:" The bounds of the arguments for these functions must be between 1 and -1, the maximum and minimum values of the sine and cosine functions, because no angle can have a sine or cosine that is greater than 1 or smaller than -1. Also it should be noted that the inverse sine/cosine function is not the same as 1/sine or 1/cosine, although the symbol makes it look like that.
That is correct: there is an error in your request. The range of the sine function spans the closed interval [-1, 1]. Thus the domain of the inverse sine function (the arc-sine) is the interval [-1,1]. However you are asking the calculator to calculate outside of the domain (7/2=3.5). If you are using the hyperbolic sine sinh, that is another matter.
Since you are familiar with sines, cosines, you know that their ranges (interval of values) varies from -1 to 1. The inverse functions of sine and cosine tkae their values in that very domain, [-1,1]. However you fed the arc sine function (sin^-1) a vlaue of (25/20.48) and that value is obviously larger outside the [-1,1] domain, hence the DOMAIN error message.
No such domain limitations exist for arc tangent (tan^-1) because the range of the tangent function spans the open interval ]negative infinity to positive infinity[.
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