# Inverse trigonometric functions

Trigonometric functions provide information about the ratio of the side lengths $a$, $b$ and $c$ in a right triangle, given an angle $x$. Now inverse trigonometric functions go the other way, and exposes the angles, given these ratios: $$\sin(x) = \frac{a}{c} \implies \arcsin\left(\frac{a}{c}\right) = x$$ $$\cos(x) = \frac{b}{c} \implies \arccos\left(\frac{b}{c}\right) = x$$ $$\tan(x) = \frac{a}{b} \implies \arctan\left(\frac{a}{b}\right) = x$$ ## Intro

Eratosthenes (born 276 B.C) is known for his estimate of the earth's circumference.

He observed that the shadow of two poles wasn't as long in Cyrene as in Alexandria, cities separated by km.

To work out the earth's circumference, he had to measure the following angle. Erastothenes was a brainy guy, but he used a rod as measurement. If he'd used inverse trigonometric functions, the answer would've been more accurate.

## Concept

Here's the unit circle: A trigonometric function may be thought of as a line from the origin to the arc. Conversely, the inverse trigonometric function brings you back to the origin.

If we feed an inverse function a value, it'll spit out an angle

Each trigonometric function has an inverse trigonometric function:

• and

• and

• and

## Math

Inverse trigonometric functions are inverse functions. This gives:

Similar relations hold for and .

Note that for several values of . If fed , doesn't know whether to return , or .

We need to specify that , otherwise will be in a confused state. In general, and are only defined on the interval .

Their older sibling is well-defined on .

## Inverse trigonometric functions

There has been a power failure. You don't own a regular watch, so you don't know what time it is.

Since you have a work interview at 11:00, you need to find out what time it is. As your neighbors are a bit strange, you would rather not ask them.

Instead, you go outside and have a look at the shadow from the tree. Let's assume the sun moves radians in hours, starting at radians at midnight. If you can work out the angle , you can estimate the time. You estimate the tree to be m tall, and the shadow to be m. But what is the angle ?

Just to recap: we know . If you use your battery-driven calculator and type in , you get radians. This is ! Since , the time is about 08:40.

is an inverse trigonometric function. It's also written , because it's the inverse function of .

Given a value produced by , then returns the angle we fed with

Similarly, there's and .

### Conditions

In order for the inverse function to be well-defined, the original function must be bijective. If, for example, we know that and , should output or ? If can be any value, there is no way to know. Then, is not bijective.

Thus, to keep things simple, just returns values within the range to . It only accepts input values between to . For example, there's no angle such that . If we feed the value , the function will throw an error! Similarly, returns values within the range to . We also need that to be between and . can return values from to , but it accepts all kinds of input values. The reason is that assumes all values from to . Hence, it's always possible to find an angle which, passed to , gives a value . ### Derivatives

Here are the derivatives for the inverse trigonometric functions. They're definitely worth memorizing.  