What is arcsin⁡, arccos⁡, arctan⁡?

Example 4.11 (Recognizing a hidden constant)

Study $g(x) = \arctan\dfrac{1 - x}{1 + x}$ on $\intoo{-1} {+\infty}$. Chain rule and a short computation:

since $(1+x)^2 + (1-x)^2 = 2 + 2x^2$. So $g' = -\arctan'$: the function $g + \arctan$ has zero derivative on the interval $\intoo{-1}{+\infty}$, hence is constant there; its value at $x = 0$ is $\arctan 1 + \arctan 0 = \frac\pi4$. Conclusion:

On $\intoo{-\infty}{-1}$ the same derivative computation holds but the constant is different ($-\frac{3\pi}4$: evaluate the limit as $x \to -\infty$). The insight: “zero derivative implies constant” is an interval-by-interval statement — exactly the subtlety exploited in Exercise 4.6 and Exercise 4.10.

Example 4.14 (Wrapping back to the principal interval)

Compute

The recipe: replace the angle by the unique angle of $\intoo{-\frac\pi2}{\frac\pi2}$ with the same tangent, i.e. subtract the right multiple of $\pi$ (the period of $\tan$). First: $\frac{3\pi}4 - \pi = -\frac\pi4$, so the answer is $-\frac\pi4$. Second: $\frac{17\pi} 5 - 3\pi = \frac{2\pi}5 \in \intoo{-\frac\pi2}{\frac\pi2}$, so the answer is $\frac{2\pi}5$. The computation is a Euclidean division of the angle by $\pi$ in disguise — and the analogous recipes for $\arcsin$ (reflect into $\intcc{-\frac\pi2}{\frac\pi2}$, period $2\pi$) and $\arccos$ (reflect into $\intcc0\pi$) drive Exercise 4.1 and the piecewise answer of Exercise 4.10.

Example 4.15 (Adding arctangents safely)

Let us prove

Two ingredients: the tangent addition formula, and — the step beginners forget — a localization of the sum. First, $\tan(u + v) = \frac{\tan u + \tan v}{1 - \tan u\tan v}$ with $u = \arctan\frac12$, $v = \arctan\frac15$ gives

Second, each of the three angles lies in $\intoo0{\frac\pi4}$ (their arguments are less than $1$), so the sum lies in $\intoo0{\frac{3\pi}4}$; the only angle there with tangent $1$ is $\frac\pi4$. Without the localization the conclusion would be “$\frac\pi4$ up to a multiple of $\pi$” — half a proof. The same two-step discipline runs Exercise 4.8 and Exercise 4.12.