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>> [music] People often talk about inverse trig functions.

Â But that's nonsense. The trig functions aren't invertible.

Â Look at y equals sine x. Sine sends different input values, like 0,

Â pi, 2 pi, 3 pi, to the same output value of 0.

Â So, how are you going to pick the inverse for 0?

Â To get around this problem, we're only going to talk about the inverses of trig

Â functions after we restrict their domain. Here I've got a graph of sine.

Â And you can see this function is not invertible.

Â It fails the horizontal line test. But, if I restrict the domain of sine to

Â just be between minus pi over 2, and pie over 2, this function is invertible.

Â And, here's its inverse, arc sine. By convention, arc sine outputs angles

Â between minus pi over 2 and pi over 2. And also by convention, arc cosine outputs

Â angles between 0 and pi. And arc tan outputs angles between minus

Â pi over 2 and pi over 2. Note that I'm calling these inverse trig

Â functions arc whatever. You know, arc cosine, arc sine.

Â One reason to call these things arc cosine or arc whatever, is because of radian

Â measure. If this is a unit circle, then the length

Â of this arc is the same as the measure of this angle, in radians.

Â That's the definition of radians. So, to say that theta is arc cosine 1

Â half, is just to say that theta is the length of the arc whose cosine is 1 half.

Â Now, what happens when we compose the inverse trig functions with trig

Â functions? Before complicating matters by thinking

Â about trig functions, let's think back to an easier example, the square root and the

Â squaring function. And just like trig functions are not

Â actually invertible, the squaring function's not invertible, because

Â multiple inputs yield the same output. So we had to define the square root to

Â pick the non-negative square root of x. And that means that if x is bigger than or

Â equal to zero, then I have the square root of x squared is equal to x.

Â But if x is negative, this is not true. The same sort of deal happens with trig

Â functions. So if theta is between minus pi over 2 and

Â pi over 2, then acrc sine of sin of theta is equal to theta.

Â But if theta's outside of this range, this is not the case.

Â If theta's ouside of this range, yes, arc sine is going to produce for me, another

Â angle, whose sine is the same as the angle theta.

Â But there's plenty of other inputs to sine which yield the same output.

Â Things are even more complicated if you mix together different kinds of trig

Â functions. For example, the sine of arc tan of x

Â happens to be equal to x divided by the square root of 1 plus x squared.

Â Where would you possibly get a formula like that?

Â Well the trick is to draw a right triangle, the correct triangle.

Â Well here I've drawn a right triangle. And I've got an angle theta.

Â And I've drawn this triangle so that theta's tangent is x.

Â That means that theta is the arc tangent of x.

Â Now by the Pythagorean theorem, I know the hypotenuse of this triangle has length of

Â square root of 1 plus x squared. And that gives me enough information to

Â compute the sine of theta. The sine of theta is this opposite side x,

Â divided by the hypotenuse, the square root of 1 plus x squared.

Â This tells me that the sine of arc tan of x must be x over the square root of 1 plus

Â x squared. Drawing pictures will get you very far in

Â understanding all of these relationships. I think it's basically impossible just to

Â memorize all of the possible formulas that relate inverse trig and trig functions.

Â But if you're ever wondering what those formulas are, you just draw the

Â appropriate picture, and then you can figure it out right away.

Â