Below is my wrong solution:

By the formula $\arccos x + \arcsin x = \pi / 2$, for $ x \in [-1, 1]$, I have

$$ y = \frac{\pi}{2} - \arcsin(\sin x) \text{ (because $\sin x \in [-1, 1]$)} \\ \implies y = \frac{\pi}{2} - x, \ \text{for $x \in R$} $$

But the correct solution in my book is enter image description here

May anyone tell me why I am wrong and how to solve the question?

  • 1
    $\begingroup$ This is because $\color{red}{\arcsin(\sin x)=x}$ does not hold for all real values of $x$. In fact, it only holds for $x\in \left[-\pi/2,\pi/2\right]$. Some more discussion can be found here for example. $\endgroup$ – Minus One-Twelfth Feb 24 '19 at 3:30
  • $\begingroup$ Here is a video that discusses how to graph $\arcsin(\sin x)$. $\endgroup$ – Minus One-Twelfth Feb 24 '19 at 3:36
  • $\begingroup$ Opps! I made a stupid mistake. $\endgroup$ – Wang Wenjun Feb 24 '19 at 3:38

The problem is that $\arcsin(x)$ is not equal to $x$ for all $x \in \mathbb R$. This is because $\arcsin$ is defined as $$\arcsin : [-1,1] \to [-\pi/2,\pi/2]$$ You can see from here that $\arcsin$ can only have values from $[-\pi/2, \pi/2]$. But, the expression $y=\frac{\pi}{2}-x$ can have any real value.

What you need to do is to somehow restrict $x$ to the interval $[-\pi/2,\pi/2]$. You do this by adding a constant, but that constant depends on the interval $I\subset \mathbb R$ that you are analyzing.

One thing I like to do with functions like these is to first differentiate and then integrate the function.

Namely, we have $$(\arccos(\sin x))' = -\frac{\cos x}{\sqrt{1-\sin^2x}} = -\frac{\cos x}{|\cos x|} = -\text{sgn}(\cos x)$$ for $\cos x \neq 0$. Then, by integrating we get $$y(x)=\arccos(\sin(x)) = -\int\text{sgn}(\cos x)\ dx = -x\cdot\text{sgn}(\cos x) +C$$ Because $\arccos'(\sin x)$ is not defined for $\cos x = 0$ i.e. $x=(k+\frac{1}{2})\pi$, $k\in \mathbb Z$, this integral is not defined in those points. Keep in mind that this constant depends on the interval $[(k+\frac{1}{2})\pi,(k+\frac{3}{2})\pi]$ that you are considering. To find the constant for each interval, use the fact that $y(x)$ is continuous and find some characteristic points that you can easily evaluate the function at. In your specific problem, determine the constant such that $$-x_k\cdot\text{sgn}(\cos x_k) +C=0$$ where $$x_k = \frac{4k+1}{2}\pi$$

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