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

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

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

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

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

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

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

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And you can see this function is not
invertible.

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

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

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

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

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

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

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

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

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

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

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

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

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

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functions?
Before complicating matters by thinking

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

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squaring function.
And just like trig functions are not

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Where would you possibly get a formula
like that?

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

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

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

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That means that theta is the arc tangent
of x.

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

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

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

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

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

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x squared.
Drawing pictures will get you very far in

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

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

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

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

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.