can anyone help me with graphing the inverse of trigonometric functions? i can't visualise them ://

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any video suggestions might help too

[errant meteor]

you mean csc, sec, cot?
or do you mean arcsin, arccos, arctan?

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csc sec cot mainly

[errant meteor]

you start off by getting familiar with the default shapes y = csc(x), y = sec(x), and y = cot(x)

I need to find a good picture of those to show

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im totally okay with that but i just can't seem to visualise taking the reflection of it using y=x

[errant meteor]

nah

[errant meteor]

youre mistaking multiplicative inverse with functional inverse

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oh

[errant meteor]

multiplicative inverse just doing 1 divided by

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yeah

[errant meteor]

that is a vertical transformation that is not easy to see

its like a heavily distorted mirror

here's an example

the red graph is sin(x)
the blue gran is 1 / sin(x), or csc(x)

keep in mind that 1/(small) = big

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yes i get that

but how do i graph the inverse of csc

[errant meteor]

I dont

what

which one

be more specific

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bro im so lost

[errant meteor]

you asked a question before I even began

[errant meteor]

bro

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this thing

[errant meteor]

youre

doing this again

thats functional inverse

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oh

[errant meteor]

lets take a brief moment

do you know what a functional inverse is

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sure

not yet

[errant meteor]

do you know what an inverse function is

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yes

[errant meteor]

🤔

look at those words closely

"functional inverse"

"inverse function"

theyre identical arent they

[umbral parcel]

oh

kinda

[errant meteor]

they mean the same thing

[umbral parcel]

ohhh

okay i get it now

what is a multiplicative inverse

[errant meteor]

man we gotta relearn how to use that word

there are three kinds of inverses that can happen

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okay

[errant meteor]

• additive inverse
• multiplicative inverse
• functional inverse

this is three different ways you can create an "opposite" version of something

now you would expect something along with its opposite would "cancel out," right?

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wait ik this

[errant meteor]

just go with common sense here, yes/no

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additive inverse is 42 and -42

[errant meteor]

yep

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multiplicative will be 42 and 1/42

[errant meteor]

yep

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right?

[errant meteor]

now we can use the same idea for functions

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okay

[errant meteor]

the additive inverse of f(x) is **-**f(x)

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yeah

[errant meteor]

so for example the additive inverse of x + 1 is -x - 1

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right

[errant meteor]

the idea is still that if you add with the additive inverse, you get 0

(x + 1) + (-x - 1) is always 0

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hmm

[errant meteor]

you can think of it as "a function that tells you the additive inverse of x + 1"

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okay

[errant meteor]

and you can see there if you add it with x + 1, you get 0

does that make sense

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yes

it does

[errant meteor]

nice, now for multiplicative inverse

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can we skip to functional if that's okay

[errant meteor]

well you really need multiplicative inverse

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oh okay

[errant meteor]

.

youre not doing functional inverses at all

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oh

sorry

[errant meteor]

now for multiplicative inverses

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yes

[errant meteor]

the idea is that if you multiply with the multiplicative inverse, you get 1

1 is like the "0" of multiplication

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yep

[errant meteor]

in general these special numbers that are in the middle of it all are called identities

so 0 is the "identity" of addition

and 1 is the "identity" of multiplication

identities do nothing when you use the operations, theyre like "empty"

you sort of get that, right

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yess

[errant meteor]

(its not related to anything but now you know)

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okay

[errant meteor]

in general,

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but like

[errant meteor]

so for sin(x),

its "multiplicative inverse" is 1 divided by sin(x)

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sin^-1 x is not the inverse of sinx right

sorry wait

[errant meteor]

we havent gotten to that yet

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okay

[errant meteor]

sin^-1 x is the functional inverse of sin x

we need to get into notation before you can bring those in

[errant meteor]

for now we at least we gotta know what these inverses are

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no like sinx isn't restricted

but sin^-1 x is

[errant meteor]

youre overstepping here

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okay sorry

i believe you

go on

[errant meteor]

please wait until we get to functional inverses before asking questions about it

[umbral parcel]

sure

[errant meteor]

you havent even gotten doing 1 divided by

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sorry

[errant meteor]

the multiplicative inverse of f(x) is 1 / f(x)
so for example the multiplicative inverse of x + 1 is 1/(x+1)

the idea is that if you multiply with the multiplicative inverse, you get 1 (the "multiplicative identity")
(x + 1) times 1/(x+1) is always 1

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yep

[errant meteor]

heres the graph

red is the original function
blue is the multiplicative inverse

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bro see this is the thing

[errant meteor]

bro

alr Ill move on

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i just can't wrap my head around how the inverse of x+1 's graph translates to that

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no no no

im with you

[errant meteor]

what did you think I was going to say

I just showed you an image of the graph

I wouldnt show you a picture of something you already knew

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right

sorry

im just

scared

[errant meteor]

of?

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finals

in 2 months

im sorry

go on

[errant meteor]

thats 2 whole months, you have plenty of time

lets try to focus on just one value

this is partly what youre looking for

lets say x = 2

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yes

[errant meteor]

x + 1's value then is 3

and 1/(x+1)'s value then is 1/3

happens to be the multiplicative inverse of 3

[umbral parcel]

rightt

[errant meteor]

another word for "multiplicative inverse" is "reciprocal"

so the reciprocal of 3 is 1/3,
the reciprocal of 1/3 is 3,
and the reciprocal of x+1 is 1/(x+1)

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yep

[errant meteor]

so if you want to mention the multiplicative inverse but you dont want to type 30 letters, we're gonna go with "reciprocal" from now on

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okay

[errant meteor]

the idea is that each inverse behaves in a uniquely different way and so have different names

anyways

say x = -3

you can see here x+1 then is -2
and the reciprocal 1/(x+1) then is -1/2

-1/2 is the reciprocal of -2

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yes

[errant meteor]

so in this way, the "reciprocal" of x+1 means you take the reciprocal of the output

(-3) + 1 = -2
1/((-3)+1) = -1/2

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right

[errant meteor]

now try looking at the graph closer

pay attention to it as if in vertical slices

and youll notice that youre always seeing this reciprocal behavior regardless of the slice you choose

also notice that the graphs cross at -1 and at 1

this is because the reciprocal of -1 is still -1
and the reciprocal of 1 is 1

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wait im lost

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where

[errant meteor]

-1 and 1 on this axis only

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ohhhh

yes

[errant meteor]

very nice

youll notice its entirely vertical

theres a reason for that

the two graphs being shown here are essentially
y = f(x)
y = **1/**f(x)
where f(x) = x + 1

so you begin at x, you calculate f to get f(x)

so if you put this in,

one of them f(x) = y

the other one **1/**f(x) = y

so the other one sets y to be the reciprocal

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okay

[errant meteor]

so "y = original and y = reciprocal" gets you two graphs

red where y = original
blue where y = reciprocal

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can you give me some examples

of values

[errant meteor]

you mean me giving you?

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wtf did i just type

[errant meteor]

lol

and you can see here thats always happening for any vertical slice

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yes

[errant meteor]

here's one

so youre only looking at where the values are on the y-axis

[errant meteor]

because thats where the original & reciprocal are being stored

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im lost again...

[errant meteor]

y

-2

-0.5

reciprocal of -2 is -0.5

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yes

[errant meteor]

do you see it now in this image

just in that slice

youre just supposed to see where the points are vertically

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yeah yeah yeah

i just tried with x= 3

and i got it

[errant meteor]

nice

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thank you you're so great

[errant meteor]

youre doing good so far

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okay back to what you were saying

[errant meteor]

now that you know what to look for, we can get to figuring out how to predict the blue shape

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okay

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nah lol let's be real

[errant meteor]

there are a variety of ways you could be doing worse

and you arent doing any of those ways

you are doing good so far

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oh

thank you

[errant meteor]

so now we'll need to take a look at the function 1/x

you can think of this as a function that turns a number into its reciprocal

for example

right

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right

[errant meteor]

the first thing is that if x > 1,

then 1/x is between 0 and 1

so numbers like 2, 3, 4, 5 have reciprocals like 1/2, 1/3, 1/4, 1/5

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yep

because they keep getting smaller

[errant meteor]

thats another pattern too

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if x<1 ?

[errant meteor]

well its more of 0 < x < 1

so if x is between 0 and 1,

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yeah

[errant meteor]

the reciprocal will be bigger than 1

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it goes to infinity

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i get you

[errant meteor]

now you compare this to the height the numbers wouldve had if they were unchanged

you can now sort of see the distortion happening

being higher than 1 --> reciprocal being between 0 and 1

being between 0 and 1 --> reciprocal being higher than 1

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yes

[errant meteor]

heres an example

you can see here that the heights still follow the same rule, right

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uhm

yes

[errant meteor]

nice

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okay

[errant meteor]

one more side to consider

youll notice reciprocals are completely unaffected by sign

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yes

[errant meteor]

so the negative side behaves much in the same way

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yess

[errant meteor]

x < -1 means that the reciprocal is between -1 and 0
-1 < x < 0 means the reciprocal is lower than -1

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wait what

[errant meteor]

x < -1 means that the reciprocal is between -1 and 0

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i get it

i was

tripping

[errant meteor]

youre making sure is all

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sorry sorry

[errant meteor]

alr try this

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okay

[errant meteor]

the blue function is the reciprocal of the red function

now you know the coordinates of the first point: (2, 4)

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okay

[errant meteor]

the second point near the bottom-right has x-coordinate 2

and is on the blue function

what is the y-coordinate of the second point?

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4

[errant meteor]

the second point has the same y-coordinate as the first point?

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wait

what

wait

[errant meteor]

that cant be it

its gotta be different

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0.25

?

[errant meteor]

very good

thats it

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oh

sure

[errant meteor]

so now we finally can get to justifying the shape of csc(x), cot(x), and sec(x)

so that you can draw them in case you forget

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yay

[errant meteor]

heres one

the original function is cos(x)

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yes

[errant meteor]

the reciprocal is called "sec(x)", which means 1/cos(x)

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right

[errant meteor]

for example cos(pi/3) = 1/2

this means that sec(pi/3) = 2

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yess

[errant meteor]

now Ill go draw two thin grey lines

theyre to show where -1 and 1 are at

should add numbers to those

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yep

[errant meteor]

notice there that cos(x) is always between -1 and 1

and so sec(x) is always at least "1 away" from 0

so sec(x) is never between -1 and 1

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at 0 sec(x) is undefined right

[errant meteor]

youre thinking of a different function

cos(0) = 1

so sec(0) = 1 also

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oh

bro im so dumb

[errant meteor]

you can see there though

cos(pi/2) = 0

and sec(pi/2) is undefined

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yes pi/2

omg

bcs at pi/2

cos is 0

sec will be 1/0

omg

[errant meteor]

yep

heres another thing

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man am i dumb

[errant meteor]

you can see why 1/0 cant be made to make sense

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yeah

[errant meteor]

the left half of pi/2 seems to indicate that 1/0 should be positive infinity

since sec(x) keeps going higher and higher as x gets closer to pi/2

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yes

[errant meteor]

however the right half of pi/2 seems to indicate that 1/0 should be negative infinity

they must both be correct if 1/0 has to equal one thing

you can see here these dont mix

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wait im lost again

[errant meteor]

the thing is

most people think 1 / 0 = infinity

but you can see above that 1 / 0 cant even decide on whether its positive or negative

you can reach 1 / 0 from either side

but either side suggest different values for 1 / 0

left side suggests 1 / 0 = positive infinity

right side suggests 1 / 0 = negative infinity

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wait what right side

and what left side

[errant meteor]

left/right side of the purple line

the purple line at x = pi/2

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wait lemme add that to my desmos

[errant meteor]

this is just to show what 1/0 is

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how do i go from numbers to pi

[errant meteor]

click the wrench in the upper-right corner

youll see an option for "Step:"

type pi/2

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yes

[errant meteor]

this will change that axis to step by pi

[umbral parcel]

ohhh

i get it now

continue

[errant meteor]

sick

now this 1/0 behavior is also to be expected

its because of what cos(x) does

from the left side, cos(x) is positive and heading towards 0
from the right side, cos(x) is negative and heading towards 0

as the reciprocal,

from the left side, sec(x) is positive and heading towards positive infinity
from the right side, sec(x) is negative and heading towards negative infinity

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yes

[errant meteor]

so each of these purple spikes appear whenever the original function is 0

each of them are called "asymptotes"

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what purple spikes

...

[errant meteor]

sorry the blue ones

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oh okay

[errant meteor]

Im calling them purple spikes because I used a purple line right through it

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right

[errant meteor]

the blue spikes indicated by the purple line are called "asymptotes"

these spikes only get to be vertical due to the way we got them

so these are called "vertical asymptotes"

asymptotes are spikes that are infinitely high
you expect to get as close to the purple line as possible from an asymptote

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ohh

[errant meteor]

you can get real close

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yess

[errant meteor]

but usually you never touch the asymptote

remember that these spikes always go out of the graph

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yea bcs it's approaching infinity

[errant meteor]

they continue forever

nice

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right

[errant meteor]

alr I think we've pointed out enough details

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tell me whenever we finish this topic bcs i have to study phy and chem too today

[errant meteor]

yea

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and i used up a lot of time for math

i hope you get it

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it's proven worthy so far tho

[errant meteor]

thanks

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you're so great

[errant meteor]

now you have a lot to remember by when you remember the default shape of these graphs

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and patient

[errant meteor]

thanks

red = sin(x)
blue = csc(x)

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right

[errant meteor]

for the record,
red = cos(x)
blue = sec(x)

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yes

[errant meteor]

this is the hardest to read
red = tan(x)
blue = cot(x)

it may be easier to consider if I stretch one of the axes

so that you have more space to consider this

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kinda better now

yeah

[errant meteor]

and this matches up with
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)

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yes

[errant meteor]

now that youre really familiar with the shapes,

do you know how to graph csc(x), sec(x), and cot(x)

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yes kinda

[errant meteor]

thats good already

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yay

[errant meteor]

theres a big catch I gotta mention when getting over to functional inverses

its an inconvenient notation to use ^-1

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oh

[errant meteor]

because where the ^-1 is written changes what it means

first, we know that the inverse function is written as f^-1 or f^-1(x)

,,f^{-1}(x)

[stable oxide]

mtt07734

[errant meteor]

this doesnt match up with the reciprocal function which is written as 1/f or 1/f(x)

,,\frac1{f(x)}

[stable oxide]

mtt07734

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oh

[errant meteor]

inverse function = functional inverse
reciprocal function = multiplicative inverse

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rightt

[errant meteor]

that ^-1 though can also act as a power

like in 5^-1 = 1/5

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yep

[errant meteor]

this is where the confusion begins in reading ^-1

so the usual expected way is:

,,f^{-1}(x)\text{ inverse function}
\f(x)^{-1}\text{ reciprocal function}

[stable oxide]

mtt07734

[umbral parcel]

yeah i get that

[errant meteor]

and also f^-1 only means inverse function

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okay

[errant meteor]

so you leave out the (x) and its inverse only

you have to write it as f(x)^-1 to mean (f(x))^-1 as you said

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right

can we like lowkey

continue this perhaps tomorrow

[errant meteor]

no guarantees

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i hate to interrupt

[errant meteor]

but you can always ask this again or try to find a video on it

[umbral parcel]

or whenever you're free

i would make some time

[errant meteor]

youre in a good position to learn about the arctrig functions

you can google those

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oh okay

[errant meteor]

those are the proper names for what the inverse trig functions are

so the inverse function to sin(x) is sin^-1(x) or arcsin(x)

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really thank you for all the help man

[errant meteor]

np

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oh okayy

😘

[errant meteor]

alr you can go figure that out on your own

so you dont need me to be here yk

[umbral parcel]

oh...

[errant meteor]

if you want to ping me though its a gamble whether Ill be there

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just text me whenever you're free i'll def make sure to find out time for this

[errant meteor]

with how busy Ill be tomorrow youll have to text me instead

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oh okay sure

[errant meteor]

alr

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:))))

byeee :3

[errant meteor]

cya