## Is it possible to ask Mathematica to give all the roots of the given function?

I have this equation

$$36 \cos \frac{3 x}{4} \cos \frac{27 x}{20} \left(\cos \frac{3 x}{5} +2 \cos \frac{21 x}{10} \right)=0$$

Is it possible to ask Mathematica to give all the roots of the function on the domain $$0? Preferably as a rational multiple of $$\pi$$? (the plot of the function is attached)

36 Cos[(3 x)/4] Cos[(27 x)/20] (Cos[(3 x)/5] + 2 Cos[(21 x)/10]) == 0  

## How to spawn an object at a given position and angle?

Let’s say I have an enemy at position ( 150, 150 ) in my screen and I want it to shoot a bullet from its position at an angle of 315° (south-east if we’re cosidering a regular unit circle)…

I’ve tried doing something like this (pseudo-code):

bullet.xpos = enemy.xpos + (cos(315) * bullet.radius )

bullet.ypos = enemy.ypos + (sin(315) * bullet.radius )

The idea is to change the bullet’s position by increasing its radius…

However, the problem with this code is that the angle at wich the bullet gets placed it’s not really the one specified (315° in this case) because of course, the final result going into the bullets position gets influenced by the enemy’s position ( bullet.xpos = enemy.xpos… and bullet.ypos = enemy.ypos… ) but I still need to consider it since it will be the starting point for the bullet.

I might be overlooking something but I’m not sure how to actually do this… Thanks for taking the time

## Is Additional Fighting Style underpowered as the only class feature at a given level?

This question is inspired by considering minor homebrew improvements to the Champion subclass, which is frequently but not universally considered a weak subclass.

Looking at the class features, the following occurred to me about Additional Fighting Style (level 10):

1. Most fighting styles don’t synergize. The main exception is taking both a defensive and offensive fighting style, leading to common recommendations of Defense as the go-to choice for level 10 Champions.
2. Every class that gets a first fighting style gets additional features at that level (e.g. Spellcasting for level 2 Paladins and Rangers).

So Additional Fighting Style by itself seems pretty weak compared to the level 10 features of other fighter subclasses. Hence the question.

## Where do I see a full, exhaustive list of the resources a given Google Cloud Platfrorm project uses?

I have just logged in to a Google Cloud Platform account, and it has several old Projects. I’d like to clean up by deleting the ones I don’t need, but for that I need to see a full list of services each Project uses.

I went to the Cloud Resource Manager, and it mentions a Compute Engine Service Agent here and there, but that’s it. I know one of those Projects must have some API Keys that I still need, but neither this page, nor the Billing page tell me which one.

Searching didn’t help. This looks like the closest, but doesn’t make any sense to me and looks overcomplicated for such a simple task.

Is there one single page that will show me everything a given Project uses, so I am informed enough to decide if I still need it? Please help!

## How can I find in mathematica if my equation have solution or not on a given interval?

I’m new to mathematica, I usually used wolfram alpha, however since the equation that I’m working with is a long equation I need to use mathematica. This is an example of the problem. So, I want to know if the equation will have solution or not when m>=4 and n>=3.41421m. I don’t know what command to use and when I enter this, it said that m>=4 is not a valid variable. What should I do?

Solve[{Binomial[n-1,2]-2[(Binomial[m-1,2])+(n-m)(m-1)-1]<=0}, {m>=4, n>=3.41421m}, Reals]

## Given the following constraints what is the maximum number of attacks in a round for 3.X edition?

A former DM has had the same recurring NPC/GMPC since I started playing in his game. This was 20+ years ago and we started in 1st edition and slowly made our way through the years and editions. We updated our characters as we went to the new editions. Now this NPC/GMPC is the most reviled in his games, any time he shows up all the players immediately want him dead. We stick to character though.

The question will be broken up to hopefully get expert answers from each of the editions in which we played in this particular question it will be specific to 3.Xe. I am skipping 4e (as we all hated it and only played one session) and 5e because I know for a fact that it is not possible there (yet).

The question is as follows:

Give the following constraints what is the maximum number of attacks in this edition:

1. NPC is an Elf (In this edition it is irrelavent I suppose).
2. He was a Thief-Acrobat in previous editions and I assume multiclassed, probably Fighter-Thief, due to how 3.X prestige works let’s assume just Rogue for this edition, although any official prestige class is fair game. Character focused on tumbling and thrown knife attacks.
3. The weapon of choice was throwing knives.
4. Assume unlimited ammunition as he had a bandalier that had the knives return.
5. I know he could throw 3 knives at a time (pretty sure this was a thing for shuriken from Oriental Adventures).
6. Assume all official sources and Dragon Magazine since the first issue are open.
7. I know of this question and assume there is a variant with knives.
8. If I recall he threw with both hands as well.
9. Assume prestige classes are open, this DM even as a player had a tendency to stack multiclass options with min/max precision but assume Rogue is the primary focus.
10. Assume Quick Draw.
11. We were always between 8th and 15th level when I met this character.
12. I do not recall spell-casting but not ruling it out entirely but main build would likely have been focused on mundane means.
13. Assume focused magical item augmentation as well, just calling it out even though the aforementioned bandolier alluded to it, but for the most part official items other than that.
14. Assume no flying invisible helpers as suggested by HeyICanChan or for that matter third party interventions or Aid Another.

The end result in game was quite literally at least 2 dozen attacks per round, perhaps more. Which I have questioned him multiple times about the build and legitimacy but he as refused to provide any answers. I know DMs do not have to justify but this, combined with a number of other things over the years has lead to distrust. I have since stopped playing his games altogether, so this is just a verification on whether I have overreacted.

This was broken into 3 questions for each of the editions.

## Can a Chwinga Grant a Given Person More Than one Charm?

So, within the Chwinga statblock, which can be found in Tomb of Annihilation and Icewind Dale: Rime of the Icemaiden, this ability appears:

Magical Gift (1/Day). The chwinga targets a humanoid it can see within 5 feet of it. The target gains a supernatural charm of the DM’s choice. See the Dungeon Master’s Guide for information on supernatural charms.

So, the question is how this interacts with the rule on simultaneous effects. Does Magical Gift linger along with the charm it bestows, or is it an instantaneous effect with the charm simply being bestowed, and thus able to be used an infinite amount of times on a given target without losing the effect of the blessings? (Yes, I am aware of the once per day limit, I’m asking if it could still be used every day to stack up effects, or possibly multiple Chwingas using it all at the same time for the same result)

## How to ask to attach the plots vertically in the given code?

I use this code

{P1 = RegionPlot[0 <= Cos[y x] <= 1/2, {y, 0, 3}, {x, 0, 5},     PlotStyle -> LightBlue,     FrameLabel -> {{Style[Rotate["F", 270 Degree], 15],        None}, {Style["y", 15], None}}, BaseStyle -> 15],   P2 = RegionPlot[0 <= Cos[-y x] <= 1/2, {y, 0, 3}, {x, 0, -5},     PlotStyle -> LightBlue,     FrameLabel -> {{Style[Rotate["G", 270 Degree], 15],        None}, {Style["y", 15], None}}, BaseStyle -> 15]}  

and the result is

How can I ask Mathematica to attach the two plots vertically as follows

## What happens to a Summoned Demon when it hasn’t been given new commands?

This assumes that you use summon greater demon (Xanathar’s Guide to Everything, p. 166), and command it to attack your enemies. This summoned Balgura succeeds in killing all enemies you ordered it to, but during this time you were drawn away from the demon, and haven’t been able to issue it a new verbal command. Keep in mind, it hasn’t succeeded in breaking the spell via it’s CHA saves.

Would a Demon be able to act on it’s own accord if it hasn’t been given any new commands, and has no creatures to attack that "attacked it during the last turn."?