Animated Solution for Mathematics - Inverse Trigonometric Functions: If $x={\sin }^{-1}(\sin 10)$ and $y={\cos }^{-1}(\cos 10)$, then $y-x$ is equal to:

Welcome, fellow traveler on the path to JEE Advanced excellence. Today, we confront a problem that separates the rote memorizers from the true masters of mathematics.

We are asked to evaluate $x={\sin }^{-1}(\sin 10)$ and $y={\cos }^{-1}(\cos 10)$, and then find the difference $y-x$.

At first glance, your intuition might scream, "The answer is 0! Just cancel them!" But I urge you to pause. In the realm of inverse trigonometry, intuition without rigor is a dangerous trap.

First, let us ground ourselves. The number $10$ is not $10$ degrees; it is $10$ radians.

To understand where this value lives, we must look at the number line through the lens of $\pi$. We know that $\pi \approx 3.1416$.

Therefore, $3\pi \approx 9.42$ and $4\pi \approx 12.56$. Our input, $10$, sits comfortably between $3\pi$ and $4\pi$. This is our anchor.

Let us focus on $x={\sin }^{-1}(\sin 10)$. The function $f(x)={\sin }^{-1}(\sin x)$ is not a simple identity; it is a beautiful, continuous zigzag wave. Its principal range is $[-\frac{\pi }{2},\frac{\pi }{2}]$.

Since $10$ is far beyond this, we must find the specific linear segment that contains $10$. Our value $10$ lies in the interval $[\frac{5\pi }{2},\frac{7\pi }{2}]$, which is approximately $[7.85,10.99]$.

Within this specific interval, the graph of ${\sin }^{-1}(\sin x)$ is a line with a slope of $-1$ that passes through the point $(3\pi ,0)$. Using the point-slope form, the equation of this segment is:

Now, let us turn to $y={\cos }^{-1}(\cos 10)$. The function $g(x)={\cos }^{-1}(\cos x)$ is also a zigzag wave, but it is non-negative, reflecting the principal range of $[0,\pi ]$.

We already know that $10$ lies between $3\pi$ and $4\pi$. In this interval, the graph of ${\cos }^{-1}(\cos x)$ is a line with a slope of $-1$ that hits the x-axis at $4\pi$.

The equation for this segment is:

We have arrived at the final stage of our journey. We have determined that $x=3\pi -10$ and $y=4\pi -10$.

The problem asks for the difference $y-x$. Let us perform the subtraction with the precision of a surgeon:

Distributing the negative sign, we get $4\pi -10-3\pi +10$. Observe the elegance of the result: the constant terms $-10$ and $+10$ vanish into thin air.

The final result is:

This problem is a masterclass in why we must respect the domain and range of inverse trigonometric functions. It teaches us to visualize the graph, identify the interval, and trust the geometry. Keep practicing, keep visualizing, and you will conquer the JEE.