Professor Dave again, let’s talk about inverse trig functions.
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 one over sine X.
If we had the quantity sine X to the negative one power, that would equal one over sine
X, which we already know is cosecant X.
Instead, when we have the negative one 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 arcsin 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 half pi to half 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 one to one, because those are the only sine
values that are possible.
The range will be negative half pi to half pi, because those are the angles we get when
we plug each possible unique sine value.
So let’s make sure we know how to evaluate these.
What is the inverse sine of root two over two?
Well, if we recall our unit circle, the angle that gives a sine of root two over two is
quarter pi, so the inverse sine of root two over two is quarter pi.
Now let’s move on to inverse cosine.
This will be the inverse of cosine X from zero to pi.
This is a different restriction than we had for sine, because sine goes from negative
one to positive one as it moves through quadrants four and one.
Cosine spans all the possible values from one to negative one as it goes through quadrants
one and two, so we take the graph of cosine X from zero 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 one to one, and the range is zero to pi.
What is the inverse cosine of negative one half?
Well that would be the angle in this interval that gives a cosine value of negative one
half, and if we remember the unit circle, we know that must be two thirds pi.
Now let’s look at inverse tangent.
Starting with tangent X, we see that we will want to restrict things to within negative
half pi and half 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 half pi to half 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 three.
We might realize that the angle that produces a tangent of root three must have a sine value
over a cosine value that equals root three, and of the common angles, that must be a third
pi, since root three over two divided by one half is the same as root three over two times
two, which equals root three.
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 arcsin, 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.