Конспект: інтегрування синуса та косинуса

Hey everyone!

Today, we are going to do some cool things about integrals of trigonometric functions.

Assuming you all all familiar with $\sin(x)$ and $\cos(x)$, some strange things will happen when you take the integral of either of them. Here is what happens:

$$\int \sin(x) \, dx = -\cos(x) + C$$ $$\int \cos(x) \, dx = \sin(x) + C$$ $$\int -\sin(x) \, dx = \cos(x) + C$$ $$\int -\cos(x) \, dx = -\sin(x) + C$$

(Here, $C$ is the constant of integration!)

So, we can easily find that the integrals of these two trig functions tend to be periodic. But why do we get that?

If we look at the graph of $\sin(x)$ or $\cos(x)$, these two functions are both like a curve bouncing back and forth around the $x$-axis. Because the slope functions would decrease when the acceleration of the function decrease, and the same thing happens if the acceleration of the function increases, $\cos(x)$, which is the derivative of $\sin(x)$, seems to move leftward by $\pi/2$.

These are just for sine and cosine functions. When it comes to functions like $\sec(x)$ or $\cot(x)$, it gets more complex, and we will discover more about that in our next exercise. Hope you enjoy it so far!

Eric

You now know what trig functions are. You now know that $\sin(30^{\circ})=\frac{1}{2}$. But what if you wanted it do it the other way around? Using the previous example, what if you were given $\frac{1}{2}$ and wanted to get an output of $30^{\circ}$? Well, this is where inverse trig functions come in. We can express this by saying $\arcsin(\frac{1}{2})=30^{\circ}$. Simply add an $arc$ in front of the function you were working with previously and the input and output are switched! Here is a link to some practice: http://cdn.kutasoftware.com/Worksheets/Geo/9-Inverse Trigonometric Ratios.pdf

Please copy and paste it for your own use. Don't forget, $\sin(\arcsin(x))=x$. Good luck!

Notice that the domain and range of inverse trig functions are simply the range and domain of their normal counterpart, respectively. However, when you graph the inverse function, make sure to truncate it so that it stays a function. For example, when graphing the $arcsin$ function, its domain will be restricted to $[\frac{-\pi}{2}, \frac{\pi}{2}]$.