# Thread: Calculating inverse trig function

1. ## Calculating inverse trig function

Hi,

I hope someone can help. How does my calculator know what the range restrictions are on a inverse trig function? I know that inverse trig functions have a one-to-many mapping, so you must restrict the range. For example, I know that for arccosin it is convention to restrict the range between 0 and pi. I know that you could very well make the range for arccos between pi and 2pi. So why does my calculator choose the solution that has the range between 0 and pi? I input arccos(√2/2) on my calculator and I get pi/4. Why doesn't my calculator choose the solution 7pi/4? Both solutions are just as correct, from my understanding, but all I want to know is WHY my calculator chooses the first solution over the second one. I hope this makes sense.

Sincerely,
Olivia

2. ## Re: Calculating inverse trig function

The calculator is programmed to evaluate functions.

3. ## Re: Calculating inverse trig function

But why is it programmed with the range restricted to 0 to pi and not pi to 2pi when I ask for the arccosin?

4. ## Re: Calculating inverse trig function

Back in the old days, the inverse trig relations had two notations ... $y = \arccos{x}$ and $y=\cos^{-1}{x}$ were expressions for the relation shown in the first graph.

To indicate a function, the expressions were denoted with a capital letter, $y = \text{Arccos}{x}$ and $y=\text{Cos}^{-1}{x}$ to indicate a range restriction to make the relation a function.

As calculators were developed that could graph functions, the ranges were restricted to graph functions and the small letter expression came to indicate the inverse trig function. (why, I don't know)

The domain of the function $\color{red}{y = \arccos{x}}$ is $-1 \le x \le 1$ ... the range is $0 \le y \le \pi$. All calculators and most computer graphing tools follow this standard. The standard graph is shown in the second graph.

If for some reason one desired the function values for the inverse cosine function to be in the interval $\pi \le y \le 2\pi$, the function could be expressed as $\color{blue}{y = 2\pi - \arccos{x}}$ to achieve that desired range ... third graph.

5. ## Re: Calculating inverse trig function

Oh okay, that makes a whole lot more sense. Thank you very much! So am I also correct to assume that the calculator standard for the range of arccosin and arctangent to be in-between negative pi/2 and pi/2?

6. ## Re: Calculating inverse trig function

the functions ...

$y = \arcsin{x}$ has domain $[-1,1]$ and range $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$

$y = \arccos{x}$ has domain $[-1,1]$ and range $\left[0, \pi \right]$

$y = \arctan{x}$ has domain $(-\infty,\infty)$ and range $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

... check your text for domain & range of the other three inverse trig functions.

7. ## Re: Calculating inverse trig function

Sounds good, thanks