DAVE FARINA: Professor Dave, again. Let's talk about inverse trig functions. 

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Professor Dave Explains 

DAVE FARINA: We know about the six trigonometric functions, and we know about inverse functions. So it's time to learn about inverse trigonometric functions. Let's recall that when trying to find the inverse of a function, like f of x equals root x, we change this into a y and then we make x and y swap places, then we solve for y. In this case, that would involve squaring both sides, which gives us y equals x squared. 

Essentially, inverse functions sequentially undo anything operating on x in the original function. Now let's try y equals sine x. If we swap the variables, how do we solve for y? 

The only way to do this is to take the inverse sine of both sides. This will cancel out the sine operating on y and we get y equals inverse sine x. We must note that inverse sine x is not the same thing as 1 over sine x. If we had the quantity sine x to the negative 1 power, that would equal 1 over sine x, which we already know is cosecant x. 

Instead, when we have the negative 1 here, it's the inverse sine function. And rather than sine x, where you plug-in an angle and get a sine value, for inverse sine x, you plug in a valid sine value and it gives you the angle that generates it. Inverse sine can also be represented this way as arcsine x. 

We should quickly note that in order for a function to have an inverse function, it must pass the horizontal line test, because when we find the inverse, we are reflecting the function across the line y equals x, and whatever we get must pass the vertical line test to be considered a function. So sine x as a whole does not have an inverse function, but we will instead look at a restricted domain for this function, from negative 1/2 pi to 1/2 pi. This section does pass the horizontal line test, and it is the inverse of this section that gives us the inverse sine function, which will look like this. 

The domain of this function will be negative 1 to 1 because those are the only sine values that are possible. The range will be negative 1/2 pi to 1/2 pi, because those are the angles we get when we plug in each possible unique sine value. So let's make sure we know how to evaluate these. 

What is the inverse sine of root 2 over 2? Well, if we recall our unit circle, the angle that gives a sign of root 2 over 2 is quarter pi. So the inverse sine of root 2 over 2 is quarter pi. 

Now let's move on to inverse cosine. This will be the inverse of cosine x from 0 to pi. This is a different restriction than we had for sine because sine goes from negative 1 to positive 1 as it moves through quadrants 4 and 1. Cosine spans all the possible values from 1 to negative 1 as it goes through quadrants 1 and 2. So we take the graph of cosine x from 0 to pi and we reflect it across the line y equals x to get the inverse cosine function. 

As we would expect, the domain is negative 1 to 1 and the range is 0 to pi. What is the inverse cosine of negative 1/2. Well, that would be the angle in this interval that gives a cosine value of negative 1/2. And if we remember the unit circle, we know that must be 2/3 pi. 

Now let's look at inverse tangent. Starting with tangent x, we see that we will want to restrict things to within negative 1/2 pi and 1/2 pi so that it passes the horizontal line test. Now, to get the inverse tangent function, we reflect across y equals x and we get this. 

Now the domain is all real numbers, while the range is negative 1/2 pi to 1/2 pi since those are the angles we get when we plug in all the possible tangent values which go towards positive and negative infinity as cosine gets very small. These are trickier to evaluate in your head, but sometimes, we can do it, like the inverse tangent of root 3. We might realize that the angle that produces a tangent of root 3 must have a sine value over a cosine value that equals root 3, and of the common angles, that must be 1/3 pi since root 3 over 2 divided by one half is the same as root 3 over 2 times 2, which equals root 3. 

Luckily, rather than having to evaluate inverse trig functions by hand, we can always just plug them into a calculator. Just make sure it's in radian mode. Plug in the sine, cosine, or tangent value and use the buttons that say things like inverse sine or arcsine to find out what the corresponding angle is. Just remember, you may get a messy answer that will be hard to express as fractions of pi, but it will be the correct answer. Let's check comprehension. 

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Thanks for watching, guys. Subscribe to my channel for more tutorials. Support me on Patreon so I can keep making content. And as always, feel free to email me, professordaveexplains@gmail.com. 

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