Understanding the Formula for sin-1(x) and sin-1(y)
The formula for the sine inverse (or arcsine) of two variables is essential in trigonometry, particularly in applications where angles are involved. The sine inverse function, denoted as sin-1(x) or arcsin(x), is the inverse of the sine function, and it provides the angle whose sine is the given number.
When dealing with two variables, such as x and y, the formula for sin-1(x) + sin-1(y) can be represented using the following identity:
Formula
sin-1(x) + sin-1(y) = sin-1(x√(1-y2) + y√(1-x2))
Conditions
This formula holds under the condition that both x and y lie within the interval [-1, 1]. This range is crucial because the sine function outputs values only within this interval, ensuring that the inverse sine function returns valid results.
Application
Such formulas are handy in various mathematical contexts, especially in calculus and physics, where calculating angles is necessary. Using this identity allows us to simplify calculations that involve the addition of arcsine functions.
Example
If you want to calculate sin-1(0.5) + sin-1(0.5), you can plug the values into the formula:
sin-1(0.5) + sin-1(0.5) = sin-1(0.5√(1-0.52) + 0.5√(1-0.52))
This will lead you to simplify further and find the resultant angle in the respective trigonometric context.