Inverse Trig Functions: Arcsin

Introduction to the inverse trig function arcsin

If I were to walk up to you on
the street and say you, please

tell me what-- so I didn't want
to write that thick --please

tell me what sine
of pi over 4 is.

And, obviously, we're assuming
we're dealing in radians.

You either have that memorized
or you would draw the

unit circle right there.

That's not the best
looking unit circle,

but you get the idea.

You'd go to pi over 4
radians, which is the

same thing as 45 degrees.

You would draw that
unit radius out.

And the sine is defined
as a y-coordinate

on the unit circle.

So you would just want to
know this value right here.

And you would
immediately say OK.

This is a 45 degrees.

Let me draw the triangle
a little bit larger.

The triangle looks like this.

This is 45.

That's 45.

This is 90.

And you can solve a
45 45 90 triangle.

The hypotenuse is 1.

This is x.

This is x.

They're going to be
the same values.

This is an isosceles
triangle, right?

Their base angles are the same.

So you say, look. x squared
plus x squared is equal to 1

squared, which is just 1.

2x squared is equal to 1.

x squared is equal to 1/2.

x is equal to the square root
of 1/2, which is one over

the square root of 2.

I can put that in rational form
by multiplying that by the

square root of 2 over 2.

And I get x is equal to the
square root of 2 over 2.

So the height here is
square root of 2 over 2.

And if you wanted to know
this distance too, it would

also be the same thing.

But we just cared
about the height.

Because the sine value,
the sine of this, is just

this height right here.

The y-coordinate.

And we got that as the
square root of 2 over 2.

This is all review.

We learned this in the
unit circle video.

But what if someone else--
Let's say on another day, I

come up to you and I say you,
please tell me what the

arcsine of the square
root of 2 over 2 is.

What is the arcsine?

And you're stumped.

You're like I know what the
sine of an angle is, but this

is some new trigonometric
function that Sal has devised.

And all you have to realize,
when they have this word arc in

front of it-- This is also
sometimes referred to

as the inverse sine.

This could have just as easily
been written as: what is

the inverse sine of the
square root of 2 over 2?

All this is asking is what
angle would I have to take the

sine of in order to get the
value square root of 2 over 2.

This is also asking what angle
would I have to take the sine

of in order to get square
root of 2 over 2.

I could rewrite either of
these statements as saying

square-- Let me do it.

I could rewrite either of these
statements as saying sine

of what is equal to the
square root of 2 over 2.

And this, I think, is a
much easier question

for you to answer.

Sine of what is square
root of 2 over 2?

Well I just figured out that
the sine of pi over 4 is

square root of 2 over 2.

So, in this case, I know that
the sine of pi over 4 is equal

to square root of 2 over 2.

So my question mark is
equal to pi over 4.

Or, I could have rewritten
this as, the arcsine-- sorry

--arcsine of the square root of
2 over 2 is equal to pi over 4.

Now you might say so, just as
review, I'm giving you a value

and I'm saying give me an angle
that gives me, when I take the

sine of that angle that
gives me that value.

But you're like hey Sal.

Look.

Let me go over here.

You're like, look
pi over 2 worked.

45 degrees worked.

But I could just keep adding
360 degrees or I could

keep just adding 2 pi.

And all of those would work
because those would all get

me to that same point of
the unit circle, right?

And you'd be correct.

And so all of those values, you
would think, would be valid

answers for this, right?

Because if you take the sine of
any of those angles-- You could

just keep adding 360 degrees.

If you take the sine of any
of them, you would get

square root of 2 over 2.

And that's a problem.

You can't have a function where
if I take the function-- I

can't have a function, f
of x, where it maps to

multiple values, right?

Where it maps to pi over 4, or
it maps to pi over 4 plus 2

pi or pi over 4 plus 4 pi.

So in order for this to be a
valid function-- In order for

the inverse sine function to
be valid, I have to

restrict its range.

And the way that-- We'll
just restrict its range to

the most natural place.

So let's restrict its range.

Actually, just as a
side note, what's its

domain restricted to?

So if I'm taking the
arcsine of something.

So if I'm taking the arcsine of
x, and I'm saying that that is

equal to theta, what's the
domain restricted to?

What are the valid values of x?

x could be equal to what?

Well if I take the sine of
any angle, I can only get

values between 1 and
negative 1, right?

So x is going to be greater
than or equal to negative 1 and

then less than or equal to 1.

That's the domain.

Now, in order to make this
a valid function, I have

to restrict the range.

The possible values.

I have to restrict the range.

Now for arcsine, the convention
is to restrict it to the

first and fourth quadrants.

To restrict the possible angles
to this area right here

along the unit circle.

So theta is restricted to being
less than or equal to pi over

2 and then greater than or
equal to minus pi over 2.

So given that, we now
understand what arcsine is.

Let's do another problem.

Clear out some space here.

Let me do another arcsine.

So let's say I were to ask you
what the arcsine of minus the

square root of 3 over 2 is.

Now you might have
that memorized.

And say, I immediately know
that sine of x, or sine

of theta is square
root of 3 over 2.

And you'd be done.

But I don't have
that memorized.

So let me just draw
my unit circle.

And when I'm dealing with
arcsine, I just have to

draw the first and fourth
quadrants of my unit circle.

That's the y-axis.

That's my x-axis.

x and y.

And where am I?

If the sine of something is
minus square root of 3 over 2,

that means the y-coordinate on
the unit circle is minus

square root of 3 over 2.

So it means we're
right about there.

So this is minus the
square root of 3 over 2.

This is where we are.

Now what angle gives me that?

Let's think about
it a little bit.

My y-coordinate is minus
square root of 3 over 2.

This is the angle.

It's going to be a negative
angle because we're going

below the x-axis in the
clockwise direction.

And to figure out-- Let me just
draw a little triangle here.

Let me pick a better
color than that.

That's a triangle.

Let me do it in
this blue color.

So let me zoom up
that triangle.

Like that.

This is theta.

That's theta.

And what's this
length right here?

Well that's the same as
the y-height, I guess

we could call it.

Which is square
root of 3 over 2.

It's a minus because
we're going down.

But let's just figure
out this angle.

And we know it's a
negative angle.

So when you see a square root
of 3 over 2, hopefully you

recognize this is a
30 60 90 triangle.

The square root of 3 over 2.

This side is 1/2.

And then, of course,
this side is 1.

Because this is a unit circle.

So its radius is 1.

So in a 30 60 90 triangle, the
side opposite to the square

root of 3 over 2 is 60 degrees.

This side over here
is 30 degrees.

So we know that our theta
is-- This is 60 degrees.

That's its magnitude.

But it's going downwards.

So it's minus 60 degrees.

So theta is equal to
minus 60 degrees.

But if we're dealing
in radians, that's

not good enough.

So we can multiply that times
100-- sorry --pi radians

for every 180 degrees.

Degrees cancel out.

And we're left with
theta is equal to minus

pi over 3 radians.

And so we can say-- We can now
make the statements that the

arcsine of minus square root
of 3 over 2 is equal to

minus pi over 3 radians.

Or we could say the inverse
sign of minus square root

of 3 over 2 is equal to
minus pi over 3 radians.

And to confirm this, let's
just-- Let me get a

little calculator out.

I put this in radian
mode already.

You can just check that.

Per second mode.

I'm in radian mode.

So I know I'm going to get,
hopefully, the right answer.

And I want to figure
out the inverse sign.

So the inverse sine-- the
second and the sine button

--of the minus square
root of 3 over 2.

It equals minus 1.04.

So it's telling me that this is
equal to minus 1.04 radians.

So pi over 3 must
be equal to 1.04.

Let's see if I can
confirm that.

So if I were to write minus pi
divided by 3, what do I get?

I get the exact same value.

So my calculator gave me the
exact same value, but it might

have not been that helpful
because my calculator doesn't

tell me that this is
minus pi over 3.