What is an inverse tangent?

[Jean in Wisc]

I took this 35 years ago. I'm no help for my son here.

 

So....ds it taking Apologia Physics, but he has not had trig. (Bad idea?)

 

He can do the math on the calculator and get the right answer, but he doesn't know what an inverse tangent is.

 

Would anyone like to try to define it for us...please talk to us as though we are 3rd graders. LOL!

 

Thanks,

Jean

[Jane in NC]

Let's say you have a function, f, which takes x to some output, say y. The inverse function, denoted f^(-1) (and please don't confuse this -1 with an exponent!), takes one back, so to speak. The inverse, applied to that value y returns to the initial number x.

 

Let's look at an example. Consider f(x) = x^2 for non-negative x values only. You know that f(3) = 3^2 = 9.

 

The inverse of f is the square root of x. That is, f^(-1)(x) = sqrt(x). So f^-1(9) = sqrt (9) = 3. You are back with the x with which you started.

 

Now let's consider tangent: f(x) = tan x. You know that f(pi/4) = tan(pi/4) = 1.

 

The inverse tangent function, tan^(-1)(x) is also denoted Arctan(x), which might be easier to use here to avoid that -1 stuff.

 

So Arctan(1) = pi/4, bringing us back to where we began in the previous step.

 

There is more to be said but I want to make sure you understand this bit first.

 

Jane

[Jean in Wisc]

Let's say you have a function, f, which takes x to some output, say y. The inverse function, denoted f^(-1) (and please don't confuse this -1 with an exponent!), takes one back, so to speak. The inverse, applied to that value y returns to the initial number x.

 

Let's look at an example. Consider f(x) = x^2 for non-negative x values only. You know that f(3) = 3^2 = 9.

 

The inverse of f is the square root of x. That is, f^(-1)(x) = sqrt(x). So f^-1(9) = sqrt (9) = 3. You are back with the x with which you started.

 

Now let's consider tangent: f(x) = tan x. You know that f(pi/4) = tan(pi/4) = 1.

 

The inverse tangent function, tan^(-1)(x) is also denoted Arctan(x), which might be easier to use here to avoid that -1 stuff.

 

So Arctan(1) = pi/4, bringing us back to where we began in the previous step.

 

There is more to be said but I want to make sure you understand this bit first.

 

Jane

 

Thank you!

 

I can follow this enough to walk through it with ds. He says he can understand it here, but he still doesn't understand enough to know why plugging the numbers into his calculator gets what he gets. He really needs to take a class in trig.

 

Math is making sense to this boy for the first time in 13 years. This is the first time ever that I've seen him question why something works in math--and wanting to know more! He did Alg II in weeks rather than months this year. I only wish his little head had gotten math-wise earlier so that we could have done precalc, too. I don't know if Chalkdust was the missing factor or if maturity was the main part of this change.

 

Ah! The struggles-to-success stories of homeschooling.

:)Jean

[Jane in NC]

I can follow this enough to walk through it with ds. He says he can understand it here, but he still doesn't understand enough to know why plugging the numbers into his calculator gets what he gets. He really needs to take a class in trig.

 

 

 

In terms of his calculator (which I assume is a TI graphing calculator):

 

If he enters tan (pi/4) =....the display reads 1.

 

If he enters (second function) tan 1 =...the display reads .785398... which is an approximation of pi/4.

 

(Remember that one is undoing what the other one does.)

 

Here the second function/tan sequence produces the arctan or inverse tan function.

 

If your son looks at the keypad on his TI, he'll see that several functions are thus paired with their inverses: ln/e, x^2/sq root, log/10^x.

[LoriM]

Jean,

 

This is one of those cases that MUS PreCalculus (formerly known as Trig) would be a good fit for your son. He could quickly finish a PreCalc course before the fall. My dd covered Trig in about 9 weeks, meeting with 3 other students twice a week in my living room. :) They all went on to Calculus in the fall, and saw dramatic improvements in their ACT math scores that fall.

 

Lori

[Jean in Wisc]

Jean,

 

This is one of those cases that MUS PreCalculus (formerly known as Trig) would be a good fit for your son. He could quickly finish a PreCalc course before the fall. My dd covered Trig in about 9 weeks, meeting with 3 other students twice a week in my living room. :) They all went on to Calculus in the fall, and saw dramatic improvements in their ACT math scores that fall.

 

Lori

 

:( Oh, how I wish we could!

 

Ds will be back to working 12-hour days as soon as he can get his senior classwork finished.

 

I'll see what he says.

 

Jean