91Ó°ÊÓ

Inverse trigonometric functions help us find the angle when we know the sine, cosine, or tangent of that angle. In this exercise, we work with the inverse sine function, denoted as \( \sin^{-1}(x) \).
Here, \( \sin^{-1}\left(\frac{2}{3}\right) \) represents the angle whose sine value is \( \frac{2}{3} \). This is crucial because it allows us to convert a trig ratio back to an angle.
Inverse trig functions are widely used in math and engineering to solve equations involving trigonometric ratios. They appear as constants when the function's input (like \( \frac{2}{3} \) in this case) is set and fixed, thus, the outcome becomes a specific angle.

Radian measure is a way of expressing angles using the radius of a circle. It's especially common in higher mathematics. Instead of degrees, angles are measured by how far their corresponding arc stretches around the circle's circumference.
Remember, there are \(2\pi\) radians in a full circle, equivalent to 360 degrees. So, 1 radian equals about 57.296 degrees.
In this problem, when calculating \( \pi - \sin^{-1}\left(\frac{2}{3}\right) \), we find the result in radians. Converting angles to radians is common in calculus and trigonometry since it simplifies many mathematical expressions and calculations.

A constant function is one where the output value doesn't change, no matter the input. It's represented mathematically as \( f(x) = C \), where \( C \) is a constant value.
In our exercise, we determined \( f(x) = \pi - \sin^{-1}\left(\frac{2}{3}\right) \) to be a constant function because \( \sin^{-1}\left(\frac{2}{3}\right) \) is a fixed number.
The graph of a constant function is a horizontal line. No matter the value of \( x \), the output is the same, which means the graph remains flat and aligns with the constant value on the y-axis. For this problem, the line intersects the y-axis at around 0.58800, the fixed value calculated from \( \pi - \sin^{-1}\left(\frac{2}{3}\right) \). This stability and simplicity make constant functions easy to analyze and graph.