7sinx=6 qurstion?

Remember the Unit Circle? "the angle whose (trig function) = x" usually has two answers in  the interval [0, 2π). You have provided one of those answers. Let's look for the other one:

Recall: sin(π-x) = sin(π)cos(x) - cos(π)sin(x) = sin(x)

For every angle, x ≠ kπ/2 where k is an odd integer, there is another angle, π-x, that has the same sine value. 

Arcsin(x) is a function that returns values in the first or fourth quadrant only, it is up to you to find the second solution in the second or third quadrant. 

You found arcsin(6/7) = 58.997, but sin(360°-58.997) is also equal to 6/7.

The two values of x in the interval 0 ≤ x < 360 where 7sin(x) = 6 in the interval 0 ≤ x < 360° is:

x₁ = 58.997° 

x₂ = 180 - 58.997 = 121.003°

There are an infinite number of angles coterminal with those two angles which also share the value of sine..

[Use your calculator to verify that sin(58.997°) and sin(121.003°) are both equal to 6/7 within roundoff error. On my calculator they are the same within 2 x 10^-39, a very small difference indeed. The difference is due solely to using a finite decimal string for the angles. The more significant digits used for the angle measure, the smaller that difference will be.]

YES. Your work and answer are correct.

 7 sin x = 6

 sin x = 6/7

 x = sin^-1(6/7)

 x = 58.997  

 You are correct

 line

 x-intercept | sin^(-1)(6/7) ≈ 1.0297  

 x = 59° (degrees)

Almost all correct. Just recall that

sinx = 6/7 has two primary answers (and infinitely many answers).

 You gave one of the primary answers.

The sin^-1 only gives you one. It is up to you to remember to find the other(s).

So, whats the other possible result? [think of your trig circle].  

Other than labeling your answer with degrees (so it's not confused with radians, which then would not be the correct answer), this is correct.