Inverse Trig Functions: Arctan Understanding the arctan or inverse tangent function.
Inverse Trig Functions: Arctan
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- 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.
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