Two values of ArcCos not showing

Question

When I try to find

θ = ArcCos[-.5]

I am expecting more than one answer: $$2 \pi/3, 4\pi/3,...$$

But Mathematica is returning only one: 2.0944 and it is not in angle format as well. Is there a way to get all the values of angles in angle format?

Changing -.5 to its rational representation $-\frac{1}{2}$ will tell Mathematica to use exact symbolic subroutines (and hence get your desired angle format). — Conor, Feb 09 '21 at 19:52
ArcCos is a multivalued, actually infinitely many, function. What do you think MMA should return? It has to choose one branch. — Daniel Huber, Feb 09 '21 at 20:02
This is the same situation as Sqrt[4] returning only one out of two possible answers. Look up "principal branch". — J. M.'s missing motivation, Feb 09 '21 at 20:19
ArcCos is a function, and functions, at least in modern mathematics, have only one value for each input. — Michael E2, Feb 10 '21 at 05:36

Conor · Answer 1 · 2021-02-10T22:49:59.380

5

you could define a new version of ArcCos[] to produce the desired behaviour

arcCos[θ_] := Solve[θ == Cos[x] , x]

edited Feb 10 '21 at 22:49

answered Feb 09 '21 at 20:29

Conor

score 4 · Accepted Answer · edited Feb 10 '21 at 22:50

4

Try the equivalent

Solve[-1/2 == Cos[θ], θ]

and you get multiple values

edited Feb 10 '21 at 22:50

Conor

answered Feb 09 '21 at 20:01

Andreas

But I am getting Solve[True, (2 [Pi])/3] – Jasmine Feb 09 '21 at 20:04
Clear[\[Theta]]; Solve[-2^(-1) == Cos[\[Theta]], \[Theta]] – Conor Feb 09 '21 at 20:06
1

$$\left{\left{\theta \to \fbox{$2 \pi c_1-\frac{2 \pi }{3}\text{ if }c_1\in \mathbb{Z}$}\right},\left{\theta \to \fbox{$2 \pi c_1+\frac{2 \pi }{3}\text{ if }c_1\in \mathbb{Z}$}\right}\right}$$ – Jasmine Feb 09 '21 at 20:09
1

It's working now... – Jasmine Feb 09 '21 at 20:10

2 Answers2