We will apply the concepts of inverse functions to the sine, cosine, and tangent functions.
Here is the extended chart of values for these functions.
We have put it into a chart instead of an arrow diagram to save space, but the information
is still the same.
Let's take a moment to focus on the sine chart.
Remember that the inverse asks the question, "Given an output value, what input value
could it have come from?"
We'll focus on a specific value.
If sin(theta) = sqrt(3)/2, what can we know about theta?
By looking at the chart, we can see that pi/3 and 2 pi/3 are both possible values of theta.
But because of the periodicity of the sine function, we know that if we add or subtract
multiples of 2 pi to these, we will get even more values.
This means that the inverse isn't going to be a function unless we restrict the range.
In this case, it turns out that the interval [-pi/2, pi/2] is the natural choice.
So when we ask what arcsin(sqrt(3)/2) is, the answer will be pi/3.
We can look at the cosine and tangent charts in the same way.
If we look at the cosine chart, the interval [0,pi] turns out to be the most natural range to
pick.
For the tangent function, we restrict ourselves to (-pi/2, pi/2).
We have grayed out a couple values because the tangent function isn't actually defined
there at +-pi/2, but those endpoints are important values to know.
To get the inverse functions, we simply swap the rows in our restricted charts.
We could also rearrange the rows to put the x values in increasing order, but we'll
leave it in this form for now.
Let's look at this graphically.
Here is the graph of the sine function.
Recall that the graph of the inverse is the graph we get when we reflect this across the
line y=x.
This will result in a curve that vibrates left and right as it goes up the y-axis.
Notice that this fails the vertical line test, so it's not a function.
But by restricting the range to the interval [-pi/2, pi/2], we will get a graph that does
represent a function.
This is the same effect as restricting the chart of values from earlier.
Looking at the cosine function, we see that the inverse has the same back and forth behavior
as the sine function, so we would also need to restrict the range.
From the picture, it makes a little more intuitive sense why we pick the range as we do.
It has to do with finding a piece of the graph that can pass the vertical line test.
There are other pieces that we could have picked, but notice how those values end up
being further away from the origin or negative.
So the choice of the range of the inverse cosine is in this sense the natural choice.
The inverse tangent function is different because it extends from negative to positive
infinity.
We still have the problem of repeated values, so we will restrict the inverse to just the
connected piece that passes through the origin.
The important idea to get out of this video is that all of these steps are really
trying to define a function that can answer the question, "Given an output value, what
input value could it have come from?"
If you keep focused on this question, you will find the inverse functions to be less
confusing.
In the next video, we're going to take a look at a geometric interpretation of the
inverse trigonometric functions.
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