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## Precalculus

### Course: Precalculus > Unit 2

Lesson 3: Inverse trigonometric functions- Intro to arcsine
- Intro to arctangent
- Intro to arccosine
- Evaluate inverse trig functions
- Restricting domains of functions to make them invertible
- Domain & range of inverse tangent function
- Using inverse trig functions with a calculator
- Inverse trigonometric functions review

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# Intro to arctangent

Sal introduces arctangent, which is the inverse function of tangent, and discusses its principal range. Created by Sal Khan.

## Want to join the conversation?

- Could someone further illustrate why arctan is restricted to QI and QIV, whereas arccos is restricted to QI and QII? I understand why arcsin is restricted to QI and QIV, and I understand why arccos is restricted to QI and QII. Why is arctan not restricted to QI, QIV
*and*QII? For that matter, why not use all four quadrants for arctan (acknowledging that neither arcsin nor arccos use QIII)?(27 votes)- If arctan's range consisted of angles in QI, QII, and QIV, then what's arctan(-1)? There's an angle in QII, namely 135 degrees, whose tangent is -1, and there's an angle in QIV, namely 315 degrees (or -45 degrees, if you prefer) whose tangent is -1. In order for arctan to be a function, arctan(-1) must have just one value, and the same has to be true for arctan(x), no matter what real number x stands for.

Restricting the range of arctan to quadrants 1 and 4 isn't the only possible way to define it, but it work well because (a) it forms an interval, provided you think of it as -90 to 90 degrees (or -pi/2 to pi/2 radians), and leave out the endpoints -90 and 90; (b) no two angles in that interval have the same tangent (so we'll never get two answers for a question like "what's arctan(-1)?"); (c) no matter what real number x you choose, some angle in that interval has x as its tangent (so a question like "what's arctan(-1)?" will always have an answer); and (d) there aren't any angles in that interval whose tangent is undefined. Of those four properties, only (b) and (c) are absolutely necessary, but (a) and (d) make for simplicity.

The interval from 90 to 270 degrees (again not including 90 and 270) would also satisfy all four properties listed above, and in theory would work just as well as -90 to 90. However, if we defined arctan(x) to always be an angle in that interval, then we'd have to say arctan(0) = 180, which wouldn't be as satisfying as arctan(0) = 0!(34 votes)

- I have been trying to practice this concept but I find it very hard to solve without the table of angules, especially with arctan exercises, since is not always so easy find a slope of sqrt(3)/3 should we be solving them without any help?(20 votes)
- For 45-90-45 and 30-90-60 triangles, I try to memorize the SIDES of the triangles, not neccesarily the values of the trig ratios. That way I only have to memorize six SIDES, not over 20 ratios. With angles outside of these set triangles, just use a calculator.(27 votes)

- do we always have to convert the answar to radians?

if yes, why? if no, what it better?(15 votes)- You will not always be required to answer in radians. Whether you answer in radians or degrees depends on what is asked of you. Neither one is better than the other. If you are doing a trig problem, and it was not specified whether to use radians or degrees, use whichever one you are more comfortable with. I personally work the problem out in both degrees and radians to check my answer.(22 votes)

- what does he mean by vanilla tangent?(5 votes)
- The figure of speech "vanilla" refers to a popular flavor of ice cream, the idea being "something is as simple as you can get." So "plain vanilla" implies you're talking about the simplest or most common version of something. Other kinds (and combinations) of ice cream get more complex, sometimes much more complex: https://en.wikipedia.org/wiki/Ice_cream

So comparing something to vanilla ice cream is a simplicity metaphor.

(clarifying edits, typos - I swear I read these things before posting :-) )(13 votes)

- I'm having a lot of trouble with this subject. Could somebody walk me through a detailed explanation of this problem; What is the principal value of sin^-1 (-1/2)?(6 votes)
- here:

look at the question in this way

sin^-1 (-1/2)=what angle of sin?(let that be theta)

now break down the question

sin theta =-1/2....now what angle will be equal to -1/2

which is - 30 degree

therefore ur answer is -30 degree(9 votes)

- At4:29, why isn't the triangle in the 2nd quadrant? Does it make a difference?(6 votes)
- That's just a convention - the principal values of arcsin and arctan are in Q1 & Q4 while the principal values of arccos are in Q1 & Q2.(5 votes)

- How did you know how to draw a -1 slope at3:48?(8 votes)
- First he needs to discover the angle of the triangle, in this case it's 45º, and then he traces the line according to the angle.(1 vote)

- How would we find something like "arctan(5)"?(3 votes)
- This kind of thing you have to put into the calculator. Maybe there are ways to do this by hand but they are long tedious calculations. It is much simpler to just use your calculator and it is much more efficient as well. Hope this helps!(5 votes)

- At4:03how does Sal know that the triangle is a 45-45-90 triangle on the premise that it has a right angle? Wouldn't it also be safe to assume that it could be a 30-60-90 triangle, since we don't know what the angle is yet?? Thanks in advance!(4 votes)
- So Sal said tan = sin/cos, but I am going to use tan = opposite / adjacent.

So now we have a right triangle, and since we are trying to salve arctan(-1) this means we want some angle x where tan(x) = -1. Well, we know tan(x) = opposite/adjacent so -1 = opp/adj. To make a fraction equal 1, what do we need? we need the numerator and denominator to be the same. of course it's -1 so one will be negative. that means our right triangle has the two legs (non hypotenuse sides) as something like like 3 and -3. It can be any number, I just chose 3 at random.

Now, if you do not know this it is good to know, if the length of each leg of a right triangle is the same then it has matching angles, aside fromt he right angle.

Does that make sense?(3 votes)

- At3:54, how does Sal know to choose the 4th quadrant rather than the second? Couldn't the second also give you a -1 slope? Also, why is the angle -45 degrees rather than 315 degrees?(4 votes)
- Sal is following a convention that he hasn't introduced yet (it will come in a subsequent video). That's the only reason (at this point) for choosing the 4th quadrant, and for measuring the angle as -45 rather than 315. So you are right: 135 (second quadrant) is an equivalent answer, at this point. The problem is that "the inverse functions of sine, cosine, and tangent,...are not really invertible." This means we must restrict their domains to make them invertible, and this is why we are obliged (by convention, or definition) to designate -45 degrees as the (principle) value of arctan(-1). This will (hopefully) make more sense as you continue to study these inverse functions. (Thanks for the time-stamp: they really help with a question like this :-)(1 vote)

## Video transcript

In the last video, I showed you
that if someone were to walk up to you and ask you what
is the arcsine-- Whoops. --arcsine of x? And so this is going to be
equal to who knows what. This is just the same thing
as saying that the sine of some angle is equal to x. And we solved it in a couple
of cases in the last example. So using the same pattern--
Let me show you this. I could have also rewritten
this as the inverse sine of x is equal to what. These are equivalent
statements. Two ways of writing the
inverse sine function. This is more-- This is the
inverse sine function. You're not taking this to
the negative 1 power. You're just saying the sine of
what-- So what question mark-- What angle is equal to x? And we did this in
the last video. So by the same pattern, if I
were to walk up to you on the street and I were to say the
tangent of-- the inverse tangent of x is equal to what? You should immediately in your
head say, oh he's just asking me-- He's just saying the
tangent of some angle is equal to x. And I just need to figure
out what that angle is. So let's do an example. So let's say I were walk
up to you on the street. There's a lot of a walking
up on a lot of streets. I would write -- And I were
to say you what is the arctangent of minus 1? Or I could have equivalently
asked you, what is the inverse tangent of minus 1? These are equivalent questions. And what you should do is you
should, in your head-- If you don't have this memorized, you
should draw the unit circle. Actually let me just do a
refresher of what tangent is even asking us. The tangent of theta-- this is
just the straight-up, vanilla, non-inverse function tangent
--that's equal to the sine of theta over the cosine of theta. And the sine of theta is the
y-value on the unit function-- on the unit circle. And the cosine of
theta is the x-value. And so if you draw a line--
Let me draw a little unit circle here. So if I have a unit
circle like that. And let's say I'm
at some angle. Let's say that's
my angle theta. And this is my y-- my
coordinates x, y. We know already that
the y-value, this is the sine of theta. Let me scroll over here. Sine of theta. And we already know that this
x-value is the cosine of theta. So what's the tangent
going to be? It's going to be this distance
divided by this distance. Or from your algebra I, this
might ring a bell, because we're starting at the origin
from the point 0, 0. This is our change in y
over our change in x. Or it's our rise over run. Or you can kind of view the
tangent of theta, or it really is, as the slope of this line. The slope. So you could write slope is
equal to the tangent of theta. So let's just bear that in mind
when we go to our example. If I'm asking you-- and I'll
rewrite it here --what is the inverse tangent of minus 1? And I'll keep rewriting it. Or the arctangent of minus 1? I'm saying what angle gives
me a slope of minus 1 on the unit circle? So let's draw the unit circle. Let's draw the unit
circle like that. Then I have my axes like that. And I want a slope of minus 1. A slope of minus 1
looks like this. If it was like that, it
would be slope of plus 1. So what angle is this? So in order to have a slope
of minus 1, this distance is the same as this distance. And you might already recognize
that this is a right angle. So these angles have
to be the same. So this has to be a
45 45 90 triangle. This is an isosceles triangle. These two have to add up to 90
and they have to be the same. So this is 45 45 90. And if you know your 45 45 90--
Actually, you don't even have to know the sides of it. In the previous video, we
saw that this is going to be-- Right here. This distance is going to be
square root of 2 over 2. So this coordinate in the
y-direction is minus square root of 2 over 2. And then this coordinate right
here on the x-direction is square root of 2 over 2 because
this length right there is that. So the square root of 2 over 2
squared plus the square root of 2 over 2 squared is
equal to 1 squared. But the important thing
to realize is this is a 45 45 90 triangle. So this angle right here is--
Well if you're just looking at the triangle by itself, you
would say that this is a 45 degree angle. But since we're going clockwise
below the x-axis, we'll call this a minus 45 degree angle. So the tangent of minus 40--
Let me write that down. So if I'm in degrees. And that tends to
be how I think. So I could write the tangent of
minus 45 degrees it equals this negative value-- minus square
root of 2 over 2 over square root of 2 over 2, which
is equal to minus 1. Or I could write the arctangent
of minus 1 is equal to minus 45 degrees. Now if we're dealing with
radians, we just have to convert this to radians. So we multiply that times--
We get pi radians for every 180 degrees. The degrees cancel out. So you have a 45 over 180. This goes four times. So this is equal to-- you
have the minus sign-- minus pi over 4 radians. So the arctangent of minus 1 is
equal to minus pi over 4 or the inverse tangent of minus 1 is
also equal to minus pi over 4. Now you could say, look. If I'm at minus pi
over 4, that's there. That's fine. This gives me a value of
minus 1 because the slope of this line is minus 1. But I can keep going
around the unit circle. I could add 2 pi to this. Maybe I could add 2 pi to this
and that would also give me-- If I took the tangent of that
angle, it would also give me minus 1. Or I could add 2 pi again and
it'll, again, give me minus 1. In fact I could go to
this point right here. And the tangent would also
give me minus 1 because the slope is right there. And like I said in the sine--
in the inverse sine video, you can't have a function that
has a 1 to many mapping. You can't-- Tangent inverse
of x can't map to a bunch of different values. It can't map to
minus pi over 4. It can't map to 3-- what it
would be? --3 pi over 4. I don't know. It would be-- I'll just
say 2 pi minus pi over 4. Or 4 pi minus pi. It can't map to all of
these different things. So I have to constrict
the range on the inverse tan function. And we'll restrict it very
similarly to the way we restricted the sine--
the inverse sine range. We're going to restrict it to
the first and fourth quadrants. So the answer to your inverse
tangent is always going to be something in these quadrants. But it can't be this
point and that point. Because a tangent function
becomes undefined at pi over 2 and at minus pi ever 2. Because your slope
goes vertical. You start dividing--
Your change in x is 0. You're dividing-- Your
cosine of theta goes to 0. So if you divide by
that, it's undefined. So your range-- So if I--
Let me write this down. So if I have an inverse tangent
of x, I'm going to-- Well, what are all the values that
the tangent can take on? So if I have the tangent of
theta is equal to x, what are all the different values
that x could take on? These are all the possible
values for the slope. And that slope can
take on anything. So x could be anywhere
between minus infinity and positive infinity. x could pretty much
take on any value. But what about theta? Well I just said it. Theta, you can only go
from minus pi over 2 all the way to pi over 2. And you can't even include pi
over 2 or minus pi over 2 because then you'd be vertical. So then you say your--
So if I'm just dealing with vanilla tangent. Not the inverse. The domain-- Well the domain of
tangent can go multiple times around, so let me not
make that statement. But if I want to do inverse
tangent so I don't have a 1 to many mapping. I want to cross
out all of these. I'm going to restrict theta, or
my range, to be greater than the minus pi over 2 and less
than positive pi over 2. And so if I restrict my range
to this right here and I exclude that point
and that point. Then I can only get one answer. When I say tangent of what
gives me a slope of minus 1? And that's the question
I'm asking right there. There's only one answer. Because if I keep-- This
one falls out of it. And obviously as I go around
and around, those fall out of that valid range for theta
that I was giving you. And then just to kind of
make sure we did it right. Our answer was pi over 4. Let's see if we get that
when we use our calculator. So the inverse tangent of
minus 1 is equal to that. Let's see if that's the same
thing as minus pi over 4. Minus pi over 4 is
equal to that. So it is minus pi over 4. But it was good that we solved
it without a calculator because it's hard to recognize
this as minus pi over 4.