3.9 Reciprocal and inverse trigonometric functions

The three basic circular functions, sine, cosine and tangent, have reciprocals called secant, cosecant and cotangent. They add no new information, since each is one divided by a familiar function, but they make many identities, equations and integrals much shorter to write. The derivative of $\tan x$, for example, is $\sec^2 x$, and the identity $1 + \tan^2\theta = \sec^2\theta$

The inverse trigonometric functions answer the opposite question: given the value of a sine, cosine or tangent, which angle produced it? They appear whenever an angle has to be found from lengths, such as the angle of a ramp from its rise and run or the angle at which a camera must point to follow a moving object. Because sine, cosine and tangent repeat their values, each inverse has to be defined with care so that it gives exactly one angle.

The reciprocal trigonometric functions

Definitions

Each reciprocal trigonometric function is one divided by one of the three basic functions. They are written $\sec$ (secant), $\operatorname{cosec}$ (cosecant, written $\csc$ on many calculators and in some books) and $\cot$ (cotangent).

The last form of $\cot\theta$, $\frac{\cos\theta}{\sin\theta}$, is the safer one to use. It also defines $\cot\theta$

Where each function is undefined

A reciprocal is undefined wherever the function in its denominator is zero. On the unit circle, $\cos\theta = 0$ at the top and bottom of the circle and $\sin\theta = 0$ at the left and right.

A geometric picture on the unit circle

The names secant and tangent come from lines drawn on the unit circle. Take a point $P$ on the unit circle at angle $\theta$ and extend the radius $OP$ until it meets the tangent to the circle at $A(1, 0)$. In the right-angled triangle $OAT$, the side $OA$

The picture explains two facts about sizes. Since $OT$ is the hypotenuse of a triangle with a side of length $1$, $|\sec\theta| \ge 1$ for every $\theta$, and in the same way $|\operatorname{cosec}\theta| \ge 1$

Exact values

Exact values of $\sec$, $\operatorname{cosec}$ and $\cot$ come straight from the exact values of $\sin$, $\cos$ and $\tan$. Find the basic value first, including its sign from the quadrant, then take the reciprocal and rationalise the denominator if needed.

Graphs of the reciprocal functions

Each reciprocal graph can be sketched from the graph of the function it comes from, using the ideas of reciprocal graphs from 2.8:

• where the original function is $0$, the reciprocal has a vertical asymptote;

• where the original function is $1$ or $-1$, the reciprocal has the same value, so the two graphs touch;

• where the original function is small and positive, the reciprocal is large and positive (and the same for negative values);

• the reciprocal has the same sign as the original function everywhere, and the same period.

The graph of y = sec x

The graph of $y = \sec x$ is built from the graph of $y = \cos x$. Where $\cos x$ has a maximum of $1$, $\sec x$

The key features of $y = \sec x$ are:

• domain: $x \in \mathbb{R}$, $x \ne \frac{\pi}{2} + k\pi$

The graph of y = cosec x

The graph of $y = \operatorname{cosec} x$ is built from $y = \sin x$ in the same way. Its asymptotes are at the zeros of $\sin x$, $x = k\pi$

Since $\sin x = \cos\left(x - \frac{\pi}{2}\right)$, the graph of $y = \operatorname{cosec} x$

The graph of y = cot x

The graph of $y = \cot x$ has asymptotes where $\sin x = 0$, at $x = k\pi$, and crosses the $x$-axis where $\cos x = 0$

The graph of $y = \cot x$ is decreasing on every branch. It has period $\pi$ (like $\tan x$), domain $x \ne k\pi$

The table collects the features of all three reciprocal graphs.

Pythagorean identities with reciprocal functions

Deriving the identities

The Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$ from 3.6 has two companions. Dividing every term by $\cos^2\theta$

Dividing instead by $\sin^2\theta$ (whenever $\sin\theta \ne 0$) gives

The pairing is easy to remember: $\tan$ goes with $\sec$ and $\cot$ goes with $\operatorname{cosec}$. In the unit circle picture, $1 + \tan^2\theta = \sec^2\theta$

Finding one ratio from another

When the value of one trigonometric ratio and the quadrant of the angle are known, the identities give every other ratio. The identity gives the size; the quadrant gives the sign.

Proving identities

To prove an identity, start from one side (usually the more complicated one) and transform it step by step into the other side. A reliable strategy with reciprocal functions is to write everything in terms of $\sin$ and $\cos$, combine fractions over a common denominator, and then use a Pythagorean identity.

Some identities are proved more neatly by using the new Pythagorean identities directly instead of converting to $\sin$ and $\cos$. Both routes are valid; choose whichever keeps the algebra shorter.

Solving equations with the Pythagorean identities

An equation that mixes $\tan\theta$ and $\sec\theta$, or $\cot\theta$ and $\operatorname{cosec}\theta$, can usually be turned into a quadratic in one function by replacing a squared term. The method is the one from 3.8 for $\sin$ and $\cos$

When the quadratic does not factorise into standard angles, a graphic display calculator (GDC) finds the principal angle and the CAST diagram or the graph's symmetry gives the rest.

Inverse trigonometric functions

Why the domain must be restricted

A function has an inverse function only if it is one-to-one: each output comes from exactly one input. The sine function is not one-to-one, since $\sin\frac{\pi}{6} = \sin\frac{5\pi}{6} = \sin\frac{13\pi}{6} = \frac{1}{2}$

To define an inverse, each function is first given a restricted domain: an interval on which it is one-to-one and still takes every one of its values exactly once. The standard choices are

• $\sin x$ on $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$

Each interval contains the acute angles $0$ to $\frac{\pi}{2}$, so the inverse gives the familiar acute angle whenever the value is positive.

The inverse trigonometric functions therefore return an angle, in radians unless degrees are stated. The principal value is the one a calculator gives: for example, $\arcsin\frac{1}{2} = \frac{\pi}{6}$, although $\frac{5\pi}{6}$

Domains and ranges

The domain of each inverse is the range of the restricted original function, and the range of each inverse is the restricted domain.

The graphs of the inverse functions

The graph of an inverse function is the reflection of the original graph in the line $y = x$. Reflecting the restricted part of $y = \sin x$ gives the graph of $y = \arcsin x$: an increasing curve from $\left(-1, -\frac{\pi}{2}\right)$

The graph of $y = \arccos x$ is the reflection of $y = \cos x$ for $0 \le x \le \pi$. It decreases from $(-1, \pi)$

The graph of $y = \arctan x$ is the reflection of $y = \tan x$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$

Both $\arcsin$ and $\arctan$ are odd functions, so $\arcsin(-x) = -\arcsin x$ and $\arctan(-x) = -\arctan x$

Exact values of inverse functions

To find an exact value such as $\arccos\left(-\frac{\sqrt{2}}{2}\right)$, ask which angle in the principal range has the given value. The related angle comes from the standard exact values; the range of the function decides the quadrant.

Composing a function with its inverse

For any function and its inverse, $f(f^{-1}(x)) = x$ for every $x$ in the domain of $f^{-1}$, and $f^{-1}(f(x)) = x$

Simplifying expressions such as sin(arccos x)

An expression such as $\sin(\arccos x)$ asks for the sine of the angle whose cosine is $x$. Give the angle a name, $\theta = \arccos x$, so that $\cos\theta = x$ with $0 \le \theta \le \pi$

Because $\theta$ is between $0$ and $\pi$, $\sin\theta \ge 0$, so the positive root is correct:

A right-angled triangle gives the same results quickly when $x > 0$. For $\theta = \arcsin x$, label the hypotenuse $1$ and the opposite side $x$; Pythagoras' theorem gives the adjacent side $\sqrt{1 - x^2}$

Reading the triangles gives the standard results:

These hold for negative $x$ as well, because $\cos$ is positive throughout the ranges of $\arcsin$ and $\arctan$, and $\sin$, $\tan$ keep the sign of $x$

The identity arcsin x + arccos x = π/2

In a right-angled triangle the two acute angles add up to $\frac{\pi}{2}$, and the sine of one is the cosine of the other. This suggests that $\arcsin x$ and $\arccos x$ always add to $\frac{\pi}{2}$

Solving equations and transforming inverse functions

An equation in which the unknown sits inside an inverse function is solved by applying the original function to both sides. Before that, check that the value on the other side lies in the range of the inverse function; if it does not, there is no solution.

Transformations from 2.11 apply to inverse trigonometric graphs in the usual way. For $y = a\arccos(bx + c) + d$, the domain comes from solving $-1 \le bx + c \le 1$

Modelling with inverse trigonometric functions

Whenever an angle depends on two lengths, an inverse trigonometric function expresses it. An angle of elevation, the angle of a slope or the direction of a camera can each be written as $\theta = \arctan\frac{\text{opposite}}{\text{adjacent}}$. The model then shows how the angle changes as one of the lengths changes, which is a relationship that the triangle on its own does not show.

When one of the lengths in the model changes over time, the angle becomes a function of time. The horizontal asymptotes of $y = \arctan x$ then have a physical meaning: the angle approaches $\pm\frac{\pi}{2}$ but can never reach it while the other length stays fixed. The rate at which such an angle changes is found with the derivative of $\arctan x$

Exam-style questions on reciprocal and inverse functions

Self-check

• To do:

  I can define $\sec\theta$, $\operatorname{cosec}\theta$ and $\cot\theta$ and find their exact values for standard angles in any quadrant.

• To do:

  I can sketch $y = \sec x$