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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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# Using inverse trig functions with a calculator

Sal discusses the appropriate way to use the calculator in order to find an angle when its tangent value is given. Created by Sal Khan.

## Want to join the conversation?

- If the restrictions for tan are the quadrants I and IV. Why would we need more information to solve?(28 votes)
- The restrictions for the inverse function of tan, the arctan, are quadrants 1 and 4. These restrictions do not apply to the original tan function. Since the question stated tan(x)=1, assuming that the value of x is restricted to -pi<x<pi would potentially remove some answers that could have been the actual value of x.

This is similar to when you take the square root of an equation on both sides e.g. (x^2 = 4) to solve for x, but neglect the value of -2 that would also be a solution to the equation. You're essentially ignoring answers that exist. Hope I helped :)(26 votes)

- The question does not seem specific enough to answer unambiguously.(6 votes)
- That is exactly why there is a choice to get more information. In cases of ambiguity, more information helps.(19 votes)

- Couldn't options C and D also be considered as answers, because the inverse tangent function will get us the answer of the ø in the first quadrant while the C option will give the second angle which in this case is in the 3rd Quadrant. So, using options C and D, we are practically able to figure out both the angles and of-course we can add or subtract 2π to get all the possible angles. So, why is the answer option B?(5 votes)
`C`

is defined as`pi - atan(θ)`

, this would put you into`QII`

, not`QIII`

. If C said`pi + atan(θ)`

, then you would be in`QIII`

.`D`

would give the right slope and angle, but the fact remains that you have an ambiguous state that makes the choice`B`

for more information the best answer.(7 votes)

- I think the big problem I am having with khan now, is that there are so so so many gaps in what is taught.

You teach how to solve the equations but none of the theory, and for someone with no math background, that is like building a house of cards with mere slices of the cards.(4 votes)- I'm having the same problem. I am an adult learner aiming to go back to school this fall and started doing Khan in order to start at a higher level math.

KA claims to be the way to go to "master" the subjects but as I've gone further through the high school math courses here, I've found that to not be the case. Like you said, so many gaps so I had to look elsewhere. Eddie Woo's youtube channel has helped me tremendously on gaining a full understanding but you have to search his channel for the topics you want. He has playlists as well, here's one someone created about trig that I've found to be really good: www.youtube.com/playlist?list=PLYb4IXcMLZTJfLeV_rqqsSsWJaa_YK_7Z

I'd be interested in hearing how you're doing now, 4 months later(2 votes)

- But the question said select all that apply, so dosen't that mean that both get more information and type arctan(1) into his calculator should work?(3 votes)
- it really depends on the situation, you can't just type in arctan(1) without knowing more because that would make pi-arctan(1) also a valid choice. Since both of these options give two different angles we will need more information in order to choose between them.(2 votes)

- At2:37, how do you know the slope of the tangent is one?(2 votes)
- It is given that tan θ = 1 in the problem.(3 votes)

- If we had restricted the original domain of tan (x) = 1 to (-π/2<x<π/2), would that have made option 4 the correct answer?(3 votes)
- Why when I type tan^-1 of some really large quantity in the calculator it write 90 ? shouldn't it never reach 90 ?(2 votes)
- The limit of the function arctan(x) is pi/2 or 90 degrees as you go towards infinity (from the left).

So while you never actually get to 90 degrees you get so close that there's barely any difference.

But yes you're right you'll never actually reach 90 degrees, your calculator just rounds to 90 degrees (something like 89.9999999999999999999 becomes 90).

You can learn more about the subject in the calculus section about limits.(3 votes)

- Is it normal for a TI-84 Plus calculator to give inaccurate answers/not run programs properly when its battery is low?(2 votes)
- If your calculator is giving incorrect values, make sure that it is in either radian or degree mode. (Whichever one the measurements are taken in.) This could be the cause of your calculator giving incorrect values. (To check, go to mode and make sure that the unit you want to use is highlighted.)(2 votes)

- Wait, what should be the Javier's answer?(2 votes)
- The second answer is correct. More information is needed to answer the original question, and Sal explains why in the video.(2 votes)

## Video transcript

Javier is calibrating sophisticated medical imaging equipment. The manual reports that the tangent of a particular angle is one. So that's saying that the tangent and let's say that that particular angle is theta is equal to one. What should Javier do to find the angle? And I encourage you to pause this video and look at these choices and think about which of these should he do to find the angle? So let's look through each of them. So the first one... well, actually, instead
of looking at the choices, let's think about what we would do to find the angle. So they're saying that the tangent of some angle is equal to one. Well one thing that you
might want to do is say okay, if we take the inverse tangent, if we take the inverse tangent, of the tangent of theta, so if we take the inverse tangent of both sides of this, we of course would get the inverse tangent of
the tangent of theta. If the domain over here is
restricted appropriately, is just going to be equal to theta, so we could say the theta is going to be the inverse tangent of one. So it might be tempting to just pick this one right over here. Type inverse tangent of
one into his calculator. So maybe this looks like the best choice. But remember, I said if we restrict the
domain right over here. if we restrict the
possible values of tangent, of theta here appropriately, then this is going to simplify to this. But there is a scenario where this does not happen. And that's if we pick
thetas that are outside of the range of the
inverse tangent function. What do I mean by that? Well it's really just based on the idea that there are multiple
angles that have... or multiple angles whose tangent is one. And let me draw that here with a unit circle here. So we draw a unit circle, so that's my x axis, that's my y axis, let me draw my unit circle here. Actually you probably don't even have to draw the unit circle, because the tangent is really much more about the slope of the ray created by the angle, than where it intersects the unit circle as would be the case with sine and cosine. So if you have.. so you could have this
angle right over here. So let's say this is a candidate theta, where the tangent of this theta is the slope of this line, and this terminal angle, the terminal ray, you could say of the angle. The other side, the initial ray, is along the positive x axis. And so you could say, okay the tangent of this theta, the tangent of this theta is one. Because the slope of this line is one. Let me scroll over a little bit. Well, so let me write it this way. So tangent theta is equal to one. But I can construct another theta whose tangent is equal to one by going all the way over here and essentially going in
the opposite direction but the slope of this line, so let's call this theta two, tangent of theta two is also going to be equal to one. And of course you could go another pi radiance and go back
to the original angle, but that's functionally the same angle in terms of where it is relative to the positive x axis, or what direction it points into, but this one is fundamentally a different angle. So we do not know, we do not have enough information just given what we've been told to know exactly which theta we're talking about, whether we're talking
about this orange theta or this mauve theta. So I would say the get more information, there are multiple angles
which fit this description.