Inverse Trig Functions

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Inverse trig functions. All the trig functions have an input that is an angle and they give an output that is a ratio. Sometimes we know the ratio and we need to find the angle measure. For this, we need the inverse trig functions, which undo the direction of the original trig functions.

They switch around what's the input and what's the output. Each trig function has its associated inverse function. One way to denote this inverse is by writing the prefix "Arc" in front of the function. So Arc sine is the inverse of sine. Arc cosine is the inverse of cosine.

And Arc tangent is the inverse of tangent. Another way of denoting this inverse is to write in power of -1 immediately following the function. So sine to the -1 (x) is the inverse of sine x. Cosign to the -1 of (x) is the inverse of cosine (x), and tangent of the -1 of x is the inverse of tangent x.

Notice that we find the inverses only of the three major trig functions, not the others. We don't have to worry about the inverses of co-tangent, secant and co-secant, the test doesn't ask about those. You need to be comfortable with both notations for the inverse trig functions. Both the negative one notation as well as the arc notation.

For the purpose of the test, we only have to concern ourselves with quadrant one angles and positive ratios. The rules get more complicated when we consider angles of other quadrants, but the test doesn't explore those issues. The basic idea is that the inverse function reverses the input-output pair of the original function.

In other words, if the cosine of K equals 0.375, then the arc cosine of .375 would equal the angle K. When we are working with angles in triangles in this context, we use degrees, not radians. So we always use degrees for triangles in this context. In the right triangle shown, which of the following is equal to the measure of angle A?

So pause the video and then we'll talk about this. Okay, well certainly it's true that the cos(A) = 15/17. And so it can't be that A would equal the cosine of something other than 15 over 17. It also is true that that ratio of 15 over 17, since that's the cosine, it would not be the sine or the tangent, so neither of those work.

It's also true that the sine of A is 8 over 17. Opposite over hypotenuse, 8 over 17. And so that means that the sine inverse of 8 over 17 would have to equal A. So B is the answer here. Here's another problem, a slightly harder problem. Pause the video, and then we'll talk about this.

Okay, so let's think about this carefully. The arcsin of d over f is just angle D because the sine of angle D is d over f, so the arcsin of d over f is just D. The arccos(d/f), well what angle has a cosine of d/f? Well the angle E. So the arccos(d/f) = E.

So that expression arcsin(d/f) + arccos(d/f), that's just angle D + angle E. And those two angles have to add up to 90 degrees. So if C is the answer. The inverse trig functions can be written with either of two different notations, either the arc notation Arcsine, Arccosine and Arctangent.

Or the power-of-negative-one notation. Sine to the negative 1, cosine to the negative 1, tangent to the negative 1. You have to recognize both of those. Inverse trig functions have inputs of ratios and outputs of angles.

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