Wrong trigonometry data

Press SIN 4 3 2ND [ANGLE] 1 3 0 2ND [ANGLE] 2 ENTER

2ND [ANGLE] 1 gives you the degrees symbol, 2ND [ANGLE] 2 gives you the minutes symbol. If you need seconds, use ALPHA ["] (on the + key).

Note that this usually won't give you an EXACT value, but only a ten-digit approximation.

Having gone over a month without a response to my query, I assume it was just a matter of angular modes.

You need to enclose the whole function in parentheses, then square it. sin^2(X) shoulld be entered as sin(x)*sin(x) or (sin(X))^2

That is simple. It is a function like any other.
Press the key marked [SIN]
Screen displays sin(
Enter the value of the angle (make sure the current angle unit is the appropriate one).
Close the right parenthesis and press [ENTER]

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[.

While the domain of the sine function is the whole of the real line, it range spans the interval [-1,1]. In other words the sine of an arbitrary angle is imperatively somewhere in the closed interval [-1,1].
So when you want to find the arc sine corresponding to a given value of a sine function, you cannot give as argument to the arc sine [sin^-1] a value that lies outside the interval [-1,1].

For your case, the argument is 60 and it is outside the permitted domain [-1,1]. That is why the calculator protests by signaling a domain error.

What is so surprising? The sine and cosine functions have ranges that span the closed interval [-1,1]. As to the tangent function, its range is the the whole real line.

This being said, you should keep in mind that there are three angle units: the degree, the radian, and the grad.
When you want to calculate the sine, cosine, or tangent of a number, that number is not just a number. It is an angle, and as an angle it has a unit.

For this reason a calculator generally has two angle units defined (degree and radian), and possibly the grad. BEFORE calculating any trigonometric function you must ascertain that the angle is in the correct unit.

Clearly, in the problem you are trying to solve angle values are given in degrees, but your calculator is configured to interpret angles values in radians.

To set the default angle unit to degree press [MODE] , then press down arrow twice to reach the line (radian, degree). Highlight the word degree and press [ENTER] to set the default angle unit to degree.

Unfortunately the TI 8xPLUS family of calculators does not have a status bar on screen to remind you of the modes, and if you do not remember to check the current unit you get errors (at least unexpected results).

However, there is a way to avoid the errors and that is to enter the unit explicitly after the angle value. To do that you press [2nd][APPS] to open the (ANGLE) menu list. Try it.

1. First item in list is the degree symbol, a raised 0
   
2. 2nd is the minute, an apostrophe
   
3. 3rd is radian, a raised r
   

By selecting the degree symbol and pressing [ENTER}, that symbol echoes on the screen just after the angle value. Closing the right parenthesis of the trigonometric function and pressing [ENTER] will give you the result in the correct unit of your choice. This can be done regardless of the default unit the calculator is configured for.

Your calculator is set in radians. Press mode key. Arrow down to 3rd line. Highlight degree and press enter. Your calculator is now in degrees, not radians. To get out of the mode screen and back to home screen, press 2nd and then mode.

for function = (sin,cos,tan,sec,cosec,cotan)
arc(function) = function^-1, calculated in the first quadrant for simplicity(0-pi/2) (0-90deg), as all functions repeat circularly
just another method of notation

http://www.mathwords.com/t/tangent_inverse.htm

Try pressing the "On" button a couple times. That should work