Understanding the Principal Value of Arcsine: The Case of arcsin(sin(3π/5))

Introduction to Arcsine and Sine Functions

The arcsine function, denoted as arcsin or sin-1, is the inverse function of the sine function. The sine function, denoted as sin, maps an angle to a ratio between the opposite side and the hypotenuse in a right triangle. The arcsin function, on the other hand, takes a value between -1 and 1 and returns the angle whose sine is that value. The principal value of arcsin is defined within the interval [-π/2, π/2]

Step-by-Step Calculation of arcsin(sin(3π/5))

The objective is to determine the principal value of arcsin(sin(3π/5)). First, we need to understand the relationship between the angle 3π/5 and the sine function.

Convert 3π/5 to another angle

We start by expressing 3π/5 in a different form to make it easier to work with. We know that:

[ frac{3π}{5} π - frac{2π}{5} ]

By using the sine subtraction identity:

[ sin(π - x) sin(x) ]

We can rewrite:

[ sinleft(frac{3π}{5}right) sinleft(π - frac{2π}{5}right) sinleft(frac{2π}{5}right) ]

Find the Principal Value

The principal value of arcsin is defined as the angle within the interval [-π/2, π/2] whose sine is equal to a given value. The value 2π/5 is approximately 0.4π, which is within the first quadrant (since 0 ≤ 2π/5 ≤ π/2).

Given that 3π/5 is greater than π/2, it lies in the second quadrant. However, the sine function is positive in both the first and second quadrants. Therefore, the principal value of:

[ arcsinleft(sinleft(frac{3π}{5}right)right) arcsinleft(sinleft(frac{2π}{5}right)right) frac{2π}{5} ]

Conclusion

The principal value of arcsin(sin(3π/5)) is thus 2π/5.

Visual Representation

Consider the angle 3π/5 in standard position. When you draw a perpendicular line from the terminal ray down to the x-axis, you create a right triangle in the second quadrant. The reference angle between the terminal ray and a portion of the x-axis is:

[ π - frac{3π}{5} frac{2π}{5} ]

This confirms that the principal value of arcsin(sin(3π/5)) is indeed 2π/5.

Additional Insights

It is important to note that while 3π/5 is not within the range of the arcsin function, the problem is transformed by using the identity properties of the sine function. The sine function is periodic and symmetric, which allows us to find an equivalent angle within the principal range for the arcsin function.

Final Answer

The principal value of arcsin(sin(3π/5)) is:

[ boxed{frac{2π}{5}} ]