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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.

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• I don't understand how to FIND the angle which has a given tangent. Around he actually starts to get to work on the problem he made up, but it has a slope of -1 so it's very easy to find the angle. What if you're asked to find the inverse tangent of √3 ? •   Sal silently uses the property that y/x=-1, hence y=-x

Try that with sqrt(3):
y/x=sqrt(3)
y=x*sqrt(3)

Solve Pythagoras but substituting y:
x² + (x*sqrt(3))² = 1
x² + x² * 3 = 1
4x² = 1
x² = 1/4 => x=1/2, y=sqrt(3)/2

You should then RECOGNIZE the "square root 3 over 2" value as a side of a 30-60-90 triangle in a unit circle. Since y>x, x is positive and arctan is also positive, it is a 60 degree or pi/3 angle.
• 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.
• at "", Why do you include the first quadrant in your restriction of Theta? Why wouldn't you restrict your range of Theta to only the fourth quadrant? • • 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)?   • 