## Inverse Trigonometric Functions

Inverse functions, when they exist, are functions that return an element to it’s original state.

For example, if you have the function $2x$, the inverse function would be $\frac{1}{2}x$, as what ever number you ‘put in’ the first one, you would get back by using the second.

Inverse Trigonometric Functions work in the same way.

Trig. Function Name Notation
$\sin{x}$ arcsin x $sin^{-1}$
$\cos{x}$ arccos x $cos^{-1}$
$\tan{x}$ arctan x $tan^{-1}$

If for example, $\cos{90} = 0$, the inverse of this, written $\mathrm{arccos} (0)$ or $\cos^{-1}{0} = 90$.

This comes in useful when you’re using trigonometry to calculate angles, you will often be left with an equation in the form, $\cos{\theta} = x$, where $\theta, x$ are numbers, and you will need to use the inverse trigonometrical functions to solve these.

Many calculators have the inverse trigonometrical function as a secondary function, – see the image below – so you’ll need to use the shift functions on your calculator. You best bet is asking your teacher if you have any problems.

Warning

When using the inverse trigonometric functions, there’s something you need to be careful of.

As you hopefully know by now, the graphs of Sine, Cosine and Tangent are periodic, ie they repeat themselves.

What this means is that there is always more than one value for any inverse trigonometrical function. Because Sine and Cosine have a period of 360°, then if you have a solution $x$, then $x + 360^{\circ}$, $x + 720^{\circ}$ etc will also be a solution.

Similarly with the inverse tangent function, except that this had period 180°. So for any solution $x$, then $x + 180^{\circ}$, $x + 360^{\circ}$, $x + 540^{\circ}$ etc will also be a solution.

Most of the time this will not matter, it’s just something that you should be aware of and able to deal with.

Example 1

You have been asked to work out the angle, $x$, of a triangle, and so far you have deduced that the $\cos{x} = \frac{\sqrt{2}}{2}$. What is the value of the angle?

Using out calculator, we key in the Inverse Cosine of the value, $cos^{-1}(\frac{\sqrt{2}}{2})$ to get our value, 45. Simple right? Now let’s try one a little harder.

Example 2

You’re working for a company designing circular parts for a machine, (interesting eh?), and you’ve been told that the ‘Tan’ of a reflex angle is $\sqrt{3}$, what is the angle?

Well working the same way we did in our last example, keying $tan^{-1}(\sqrt{3})$ into our calculator gives us 60°, so we’re done right? Well not quite.

Looking back at the question, we are asked for a reflex angle, and as you hopefully remember from the article on angles, a reflex angle is one greater than 180° but less than 360°.

However, we also know that because the tan function repeats with a period of 180, then $\tan{x} = \tan{(x + 180)}$, so if we add 180° onto our answer, we get the real answer of 240°. You can always do a quick check on your answer by making sure that $\tan{240} = \sqrt{3}$, which it does.