Trigonometry/Inverse Trigonometric Functions

The Basic Idea
In the equation
 * $$y=\sin(x)$$

if we are given $$x$$ we can work out $$y$$.

If we are given $$y$$, can we work out $$x$$ ?

Look at the graph of $$\sin(x)$$ and you will see that for $$y>1$$ or $$y<-1$$ there aren't any answers at all, whilst for other values of $$y$$ there are infinitely many answers.

If
 * $$y=\sin(x)$$

we write the inverse of $$\sin$$ as follows
 * $$x=\sin^{-1}(y)$$

Notice the strange notation. It's just a convention, and does not really fit in well with the convention of $$\sin^2(x)$$ meaning $$\sin(x)^2$$. However, we are stuck with this notation. It is so widely used and so familiar in mathematical work that in practice it does not cause confusion. On many calculators the alternative notation $$\arcsin$$ is used.

A calculator will only give one answer (or error) for $$\sin^{-1}(y)$$. You can get some of the other answers by adding or subtracting multiples of 360°. In a later section we spell out the conventions as to which answer is given by $$\sin^{-1}(y)$$.

arcsin
A common notation used for the inverse functions is the "arcfunction" notation, prefixing the function name by "arc" or sometimes just "a".
 * $$\sin^{-1}(x)=\arcsin(x)=\mbox{asin}(x)$$ ,
 * $$\cos^{-1}(x)=\arccos(x)=\mbox{acos}(x)$$, and
 * $$\tan^{-1}(x)=\arctan(x)=\mbox{atan}(x)$$.

The arcfunctions might perhaps be so named because of the relationship between radian measure of angles and arclength--the arcfunctions yield arc lengths on a unit circle.

The Inverse Functions, Domain and Range
The inverse of sine or cosine for some values has multiple answers and some values does not exist at all. We'd like it if the inverse were a function, but according to the mathematical definition of a function it is not:


 * A function is something that given one value always gives back a unique value as its 'answer'.

$$\sin(x)$$ is a function because given any x it gives back some value. There is not really a function that is an inverse for $$\sin(x)$$ because, for example, $$\sin(20^\circ)$$ and $$\sin(160^\circ)$$ have the same value. The inverse 'function' does not know whether to go back to 20° or 160°.

To deal with this we need some more mathematical language. We have mathematical terminology for what a function operates on and where its values end up. A function like $$\sqrt{x}$$ needs to be accompanied by some agreement as to what values it can operate on and where they end up. By convention $$\sqrt{25}=5$$, though $$x=-5$$ is also a valid solution to $$x^2=25$$. So how do we describe this kind of thing?


 * The domain of a function is the set of values which it is defined on.

The function $$\frac{1}{x}$$ is defined on all values of x except for 0. Its domain is the real numbers excluding zero.

The factorial function, operates on the positive whole numbers. As a function f - $$f(1)=1\ ,\ f(2)=2\ ,\ f(3)=6\ ,\ f(4)=24\dots$$. The factorial function is usually written by writing an exclamation point after a number. So, 3 factorial is usually written as 3! and it is $$3\times 2\times 1=6$$. 4 factorial is written as 4! and it is $$4\times 3\times 2\times 1=24$$.

The domain of the factorial function is the positive whole numbers.


 * The range of a function is the set of values which it can take. The range will depend on the domain too.

Consider the function
 * $$f(x)=x^2$$

If we use ordinary numbers for x like 37.2 or -1001.56 we always find $$f(x)$$ is a positive number (or zero). The range of $$f(x)$$ is the numbers greater than or equal to zero.

Consider the function
 * $$f(x)=2x$$

If we choose the domain of $$f(x)$$ to be the numbers greater than or equal to 1, then the range of $$f(x)$$ is the numbers greater than or equal to two.

More Notation

 * The integers are whole numbers like 1,2,3,4 and also include 0, -1, -2,-3... A symbol we use to indicate the integers is $$\Z$$ . The statement $$x\in\Z$$ means exactly the same thing as x is an integer.


 * The reals include all the integers, and also fractions, and also other numbers like pi, and square root of 2. The reals 'fill in the gaps' between the integers. You'll get a better definition of the reals in a later book. A symbol we use to indicate the reals is $$\R$$ . The statement $$x\in\R$$ means exactly the same thing as x is a real number.


 * A range of numbers in the reals can often be written using interval notation. Here is an example for numbers greater than or equal to zero and less than one: [0,1). The square and round brackets have special meaning. The square bracket means the number is included. A round bracket means the number is excluded.


 * The notation (1.3,100.7] means all numbers between 1.3 and 100.7, including 100.7 but excluding 1.3.
 * The notation [-59.1,12.5] means all numbers between -59.1 and 12.5, including -59.1 and including 12.5.
 * The notation (0,71.2) means all numbers between 0 and 71.2, excluding 0 and excluding 71.2.


 * Function composition is applying one function after another. If we have two functions f and g, we write the composite function, applied to x as: $$f\circ g(x)$$ . This means first apply g to x, then apply f to that result. We can regard $$f\circ g$$ as a new function in its own right.


 * An inverse to a function $$f$$ is written $$f^{-1}$$ . If $$f$$ is the function that adds 12 to a number, then $$f^{-1}$$ is the function that subtracts 12 from a number.

Inverses to Trig functions
Some textbooks 'solve' the problem of inverses for trig functions by defining new functions $$\mbox{Sin}(x)$$, $$\mbox{Cos}(x)$$ , and $$\mbox{Tan}(x)$$ (all with initial capitals) to equal the original functions but with restricted domain.

With suitably restricted domain the functions $$\mbox{Sin}(x)$$, $$\mbox{Cos}(x)$$ , and $$\mbox{Tan}(x)$$ (all with initial capitals) do have inverses which are functions too.

The restrictions to allow the inverses to be functions are standard. Here (and for the rest of this page) we are using radians to measure angles rather than degrees:
 * $$\mbox{Sin}(x)$$ has domain $$\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$$


 * $$\mbox{Cos}(x)$$ has domain $$[0,\pi]$$ ; and


 * $$\mbox{Tan}(x)$$ has domain $$\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$$

For each function, the restricted domain includes first-quadrant angles as well as an adjacent quadrant.

Inverses, Really?
It is important to note that because of these restricted domains, the inverse trigonometric functions do not necessarily behave as one might expect an inverse function to behave. While
 * $$\mbox{Sin}^{-1}\left(\sin\left(\tfrac{\pi}{6}\right)\right)=\mbox{Sin}^{-1}\left(\tfrac{1}{2}\right)=\tfrac{\pi}{6}$$ (following the expected $$\mbox{Sin}^{-1}\big(\sin(x)\big)=x$$),
 * $$\mbox{Sin}^{-1}\left(\sin\left(\tfrac{5\pi}{6}\right)\right)=\mbox{Sin}^{-1}\left(\tfrac{1}{2}\right)=\tfrac{\pi}{6}$$.

For the inverse trigonometric functions, $$f^{-1}\circ f(x)=x$$ only when $$x$$ is in the range of the inverse function.

The other direction, however: $$f\circ f^{-1}(x)=x$$ for all $$x$$ to which we can apply the inverse function.

The Inverse Relations
'For completeness', here are definitions of the inverse trigonometric relations based on the inverse trigonometric functions. This section is really mostly about using mathematical notation to express how adding multiples of 360o to an angle gives us another solution for the inverse. Because we are working in radians we're adding multiples of two pi. Some more notation being used here:
 * Curly braces like so {} mean the 'set of', i.e. the set of whatever is inside them.
 * The symbol $$\cup$$ is read as union. The union of two sets is everything that is in either of them.


 * $$\sin^{-1}(x)=\left\{\mbox{Sin}^{-1}(x)+2\pi k\big|k\in\Z\right\}\cup\left\{\pi-\mbox{Sin}^{-1}(x)+2\pi k\big|k\in\Z\right\}$$ (the sine function has period $$2\pi$$, but within any given period may have two solutions and $$\sin(x)=\sin(\pi-x)$$)
 * $$\cos^{-1}(x)=\left\{\pm\mbox{Cos}^{-1}(x)+2\pi k\big|k\in\Z\right\}$$ (the cosine function has period $$2\pi$$, but within any given period may have two solutions and cosine is even: $$\cos(x)=\cos(-x)$$)
 * $$\tan^{-1}(x)=\left\{\mbox{Tan}^{-1}(x)+\pi k\big|k\in\Z\right\}$$ (the tangent function has period $$\pi$$ and is one-to-one within any given period)