-sin%5E(-1)(8)%2F(x)%2Bsin%5E(-1)(15)%2F(x)%3D(pi)%2F(2).jpeg "(v) Sin^(-1)(8)/(x)+sin^(-1)(15)/(x)=(pi)/(2)")
Solving the Inverse Sine Equation
In this article, we will explore the solution to the equation:
$\sin^{-1}\left(\frac{8}{x}\right) + \sin^{-1}\left(\frac{15}{x}\right) = \frac{\pi}{2}$
Understanding the Equation
The equation involves the inverse sine function, denoted by $\sin^{-1}(x)$, which is also known as the arcsine function. The inverse sine function returns the angle in radians whose sine is the input value.
In this equation, we have two inverse sine functions with arguments $\frac{8}{x}$ and $\frac{15}{x}$. Our goal is to solve for $x$.
Using the Property of Inverse Sine
One of the properties of the inverse sine function is that $\sin^{-1}(x) + \sin^{-1}(y) = \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)$. This property can be used to simplify the given equation.
Simplifying the Equation
Using the property of inverse sine, we can rewrite the equation as:
$\sin^{-1}\left(\frac{8\sqrt{1-(15/x)^2} + 15\sqrt{1-(8/x)^2}}{x}\right) = \frac{\pi}{2}$
Evaluating the Expression
To evaluate the expression inside the inverse sine function, we need to simplify the numerator.
$\frac{8\sqrt{1-(15/x)^2} + 15\sqrt{1-(8/x)^2}}{x} = \frac{8\sqrt{x^2-225} + 15\sqrt{x^2-64}}{x^2}$
Now, we can rewrite the equation as:
$\sin^{-1}\left(\frac{8\sqrt{x^2-225} + 15\sqrt{x^2-64}}{x^2}\right) = \frac{\pi}{2}$
Solving for x
Since the inverse sine function returns an angle in radians, we know that the expression inside the inverse sine function must be equal to 1, which is the sine of $\frac{\pi}{2}$.
$\frac{8\sqrt{x^2-225} + 15\sqrt{x^2-64}}{x^2} = 1$
Solving for $x$, we get:
$x^2 = 289 \Rightarrow x = \pm17$
Conclusion
In this article, we have solved the equation $\sin^{-1}\left(\frac{8}{x}\right) + \sin^{-1}\left(\frac{15}{x}\right) = \frac{\pi}{2}$ using the property of inverse sine and simplification techniques. We have found that the solutions to the equation are $x = 17$ and $x = -17$.
