Обернені тригонометричні функції

Try to remember what an inverse function is! We've seen before that an inverse function is when we reflect the function over the line $y=x$.

Let's think about what happens if we had $f(x) = \sin(x)$. We can define $\sin^{-1}(x)$ as the inverse of our function. Essentially, since $\sin(x)$ takes in an angle measure and returns a ratio of sides, the inverse function will take in a ratio of sides and return an angle! So since $\sin{\frac{\pi}{6}} = \frac{1}{2}$, we have $\sin^{-1}{\frac{1}{2}} = \frac{\pi}{6}$.

We can define the inverse functions of $\cos(x)$ and $\tan(x)$ similarly.

However, simply defining the inverse trigonometric functions like this is not enough. For example, think about whether $\sin^{-1}{\frac{1}{2}}$ equals $\frac{\pi}{6}$ or $\frac{5\pi}{6}$. We must, therefore, define the inverse function such that it returns only one value (and thus is a function).

We know that on the unit circle, the values of $\sin(x)$ are all different along the interval $[ -\frac{\pi}{2},\frac{\pi}{2}]$. Therefore, we can define the inverse function there, since then there is a one-to-one correspondence between an angle and a ratio of sides. By this definition, $\sin^{-1}{\frac{1}{2}}$ would equal $\frac{\pi}{6}$ rather than $\frac{5\pi}{6}$.

Similarly, we can define $\cos^{-1}(x)$ along the interval $[0,\pi]$, and $\tan^{-1}(x)$ along the interval $[ -\frac{\pi}{2},\frac{\pi}{2}]$.

Trigonometric Inverses

Now that we understand the basic trig functions, we can move forward to inverses of trig functions.

If we are given the sides of a triangle, and we need to find the degree of the angle, we can use the inverse trigonometric functions to evaluate! You can use these skills to solve trig equations.

The inverse of the sine function is arcsine. The inverse of the cosine function is arcsine. The inverse of the tangent function is arctangent.

Inverse Trig Functions

by Professor Dave Explains

In this video by Professor Dave Explains, you will learn about inverse trig functions.

Summary

Professor Dave introduces the concept of inverse trig functions: arcsine, arccosine, and arctangent. These are used to find an angle when given a trig ratio. Because sine, cosine, and tangent are not one-to-one over their entire domains, we restrict their domains so the inverse is a proper function.

For example, $\text{arcsin}(x)$ is defined on $[−1,1]$ and returns angles in $[−π/2,π/2]$, while $\text{arccos(x)}$ returns values in $[0,π]$, and $\text{arctan}(x)$ returns values in $(−π/2,π/2)$.