3.9 Reciprocal and inverse trigonometric functionsIB Maths: Analysis and Approaches HL: Revision notes

The three reciprocal ratios are $\sec\theta=\frac{1}{\cos\theta},\qquad\operatorname{cosec}\theta=\frac{1}{\sin\theta},\qquad\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}.$ Each has the same sign as the ratio it comes from, so use the quadrant to fix signs. If $\sin\theta=\frac{3}{5}$ and $\theta$ is obtuse, then $\cos\theta=-\frac{4}{5}$, $\sec\theta=-\frac{5}{4}$, $\operatorname{cosec}\theta=\frac{5}{3}$ and $\cot\theta=-\frac{4}{3}$.

Each reciprocal is undefined where its parent is zero: $\sec\theta$ where $\cos\theta=0$, $\operatorname{cosec}\theta$ and $\cot\theta$ where $\sin\theta=0$.

Divide $\sin^{2}\theta+\cos^{2}\theta=1$ by $\cos^{2}\theta$ and by $\sin^{2}\theta$: $1+\tan^{2}\theta=\sec^{2}\theta,\qquad1+\cot^{2}\theta=\operatorname{cosec}^{2}\theta.$ Use them to turn an equation into a quadratic in one ratio. For example $\sec^{2}\theta-\tan\theta=3$ becomes $\tan^{2}\theta-\tan\theta-2=0$, so $\tan\theta=2$ or $-1$.

They also help find extreme values: $\sec^{2}\theta-\tan\theta=\left(\tan\theta-\frac{1}{2}\right)^{2}+\frac{3}{4}\ge\frac{3}{4}$.

$\sin$, $\cos$ and $\tan$ are many-to-one, so their domains are restricted to make them one-to-one before inverting:

• $y=\arcsin x$: domain $-1\le x\le1$, range $-\frac{\pi}{2}\le y\le\frac{\pi}{2}$.
• $y=\arccos x$: domain $-1\le x\le1$, range $0\le y\le\pi$.
• $y=\arctan x$: domain $x\in\mathbb{R}$, range $-\frac{\pi}{2}<y<\frac{\pi}{2}$ (strict).

The values returned are principal values. For example $\arcsin\left(-\frac{\sqrt{3}}{2}\right)=-\frac{\pi}{3}$ (not $\frac{4\pi}{3}$) and $\arccos\left(-\frac{1}{2}\right)=\frac{2\pi}{3}$.

Described in words:

• $y=\arcsin x$ rises from $\left(-1,-\frac{\pi}{2}\right)$ through the origin to $\left(1,\frac{\pi}{2}\right)$.
• $y=\arccos x$ falls from $(-1,\pi)$ through $\left(0,\frac{\pi}{2}\right)$ to $(1,0)$.
• $y=\arctan x$ increases through the origin with horizontal asymptotes $y=\pm\frac{\pi}{2}$.

Each is the reflection in $y=x$ of the restricted $\sin$, $\cos$ or $\tan$ graph.

For a transformed function, work on the input to find the domain and on the output for the range. $y=\arccos(2x-3)$ needs $-1\le2x-3\le1$, so $1\le x\le2$, with range $[0,\pi]$. $y=2\arctan x+\frac{\pi}{2}$ has range $-\frac{\pi}{2}<y<\frac{3\pi}{2}$.

Angles of elevation and slopes often lead to $\theta=\arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)$. For a drone $h$ m above a point $40$ m away, $\theta=\arctan\frac{h}{40}$: $\theta\to\frac{\pi}{2}$ as $h\to\infty$ but never reaches it.

A reciprocal ratio can then be found exactly: $\cot\theta=\frac{40}{h}$, so $\operatorname{cosec}\theta=\sqrt{1+\frac{1600}{h^{2}}}=\frac{\sqrt{h^{2}+1600}}{h}$, positive because $\theta$ is acute.