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## Precalculus

### Course: Precalculus>Unit 2

Lesson 3: Inverse trigonometric functions

# 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)?
• 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!
• 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?
• 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.
• do we always have to convert the answar to radians?
if yes, why? if no, what it better?
• 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.
• what does he mean by vanilla tangent?
• 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 :-) )
• 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)?
• 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
• At , why isn't the triangle in the 2nd quadrant? Does it make a difference?
• 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.
• How did you know how to draw a -1 slope at ?
• 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)"?
• 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!
• At how 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!
• 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?