To solve for x in a triangle, when would you use sin and when would you use sin-1? Give an example for each type of situation.
I think by x you mean one of the angles of the triangle.
If we look at the relation y = sin x , the independent variable is x and the dependent variable is y. The value of sin x for an angle x is dependent on the value of x. sin x cannot be used to find the value of x.
If y = sin x, you can find the value of x using the relation x = arc sin y = arc sin (sin x). For x = arc sin y, the independent variable is y which is sine of the angle x. The angle x is the dependent variable here.
For example consider a right triangle with an angle equal to x, the side opposite x is of length 5 cm and the hypotenuse has a length 10. Here sin x = 5/ 10 = 0.5
If it is known that sin x = 0.5, x = arc sin (0.5) = 30 degrees.