geometry-and-trigonometry

[solid needle]

can you see that i have labeled the y coords of each of these points

based on our definition of sin?

we don't care about the x coord, cause that's cos

we ignore that for now

[amber hollow]

yeah i understand this

[solid needle]

how is the blue sin t and the red sin(-t) related?

like suppose sin t = 0.7

what would sin(-t) be

[amber hollow]

-0.7

[solid needle]

bam

so the negative sin t

gets us to sin (-t)

sin (-t) = -sin t

proven

[amber hollow]

oh that actually makes sense

[solid needle]

as does any math when it clicks

you wanna try simplifying one on your own?

[amber hollow]

well i just kinda noticed they have the same x coordinate dont they?

[solid needle]

keep going....

[amber hollow]

so if you flip on the x axis that means that you're changing sin from positive to negative. so i'm guessing it works the same way for cos but by flipping on the y axis instead?
so cos(-t) = -cos(t)?

[solid needle]

not quite

the flip on the x axis is what happens when you go from t to -t

the ANGLE is what is being flipped here

and then we reasoned that the y coord also flips

but does the x coord also flip?

[amber hollow]

no it doesnt

[solid needle]

so how would you fix that equation

[amber hollow]

cos(-t) = cos(t)?

[solid needle]

bam

proven

[amber hollow]

what if you flip the angle the other way, what are you flipping then?

[solid needle]

well let's see

let's pretend that the angle on the right is 50 deg

what is the angle on the left

assuming that it's flipped over the y axis

[amber hollow]

-50 deg...?

or does it not go negative cause ur going in the same direction?

[solid needle]

so remember where 0 is

remember that going negative is going clockwise

[amber hollow]

is it 360 - 50?

[solid needle]

well 360 is just 0

[amber hollow]

oh

[solid needle]

let's narrow it down

the angle has to be between what two values

[amber hollow]

0 and 180

so 180 - 50?

[solid needle]

there you go

ok what if the right angle was 20 instead of 50

what's the left angle

[amber hollow]

180 - 20, 160

[solid needle]

and finally

if the right angle was t

what's the left angle

[amber hollow]

180 - t

[solid needle]

perfect

so 180 - t is transforming the t by flipping it over the y axis

now small note here

we are using degrees right now

make sure that whenever you are using degrees, you are explicitly specifying degrees

either by writing it somewhere clearly or more simply using the degree symbol

it is highly important that you don't forget this

because degrees are actually a very unnatural unit to use when measuring angles

have you heard of radians?

[amber hollow]

yes, i have to use those

[solid needle]

yeah so

did you ever wonder why there 360 degrees in a full circle?

why 360?

why not 100? or 400? or 63013?

[amber hollow]

something to do with pi?

[solid needle]

nope

it's entirely arbitrary

[amber hollow]

wha

[solid needle]

like a right angle is just 1/4th the circle, right?

so if full circle = 360 degrees, then right angle = 360/4 = 90 deg

yeah?

[amber hollow]

yeah

[solid needle]

so if a full circle is 100 deg

a right angle would just be 25 deg

right?

[amber hollow]

yes

[solid needle]

feels weird because that's not what you're used to

but it works perfectly fine, no?

same with inches/cm

[amber hollow]

oh

so what makes radians so special

[solid needle]

short answer: radians are mathematically "natural"

longer answer:

so we only talked about how to geometrically understand what sin/cos are

but we have calculators that can compute this value to many decimal places

how do these calculators do that?

in calculus, maybe you've heard of taylor series?

if not, don't sweat it

basically, we can turn the sin function into an infinitely long polynomial

and the more terms we calculate, the closer it will be to the true value

[amber hollow]

the sin function?

[solid needle]

actually any trig function

im just using sin as an example

[amber hollow]

what do you mean by the function part i mean. like opp/hyp?

[solid needle]

yeah like

we only know how to understand what sin means

geometrically

but let's say i wanted you to calculate sin(57 deg)

to 10 decimal places

you'd be totally done for, right?

like how would you even begin to do that

no matter how accurately you try to draw a triangle and measure it, it wouldn't be good enough

if you drew a perfect triangle the size of the earth, you'd need to be accurate to a like a mm

it's totally impossible

[amber hollow]

yeah i see

[solid needle]

so that's why we get the calculator to compute it

but how does the calculator do it then?

it's not magic, it's just calculus

it's that ugly polynomial i posted a picture of

like i said, it's actually infinitely long

and there is a simple pattern you can use to keep going as far as you like

you can stop anywhere you want

once you have the level of accuracy you want

that part make sense?

[amber hollow]

i think so

[solid needle]

ok so

this is why radians are important

it turns out that

if you try to stick a degree value in there

like if you take that polynomial and plug in x = 57

you will NOT get sin(57 deg)

it ONLY works if you plug in the angle measure in radians

and this is basically true for almost any mathematical formula involving angles

basically all math formulas involving angles, if you're plugging raw numbers into it, you need radians

so everything eventually has to be converted back to radians before it can really be used anywhere

[amber hollow]

is it just a unit of measurement that all math formulas agreed to use

[solid needle]

it's a unit of measurement that all math formulas only work with "naturally"

you can then modify any of these formulas to work with say degrees

but it requires a very arbitrary extra step

it's not so much that math "agreed" to use radians

it turns out this is just how the math happens

it's like saying that math didn't just "agree" to have numbers go even odd even odd even odd etc

it just happens to be that way

[amber hollow]

huh all right

[solid needle]

so make sure to be extra careful when dealing with angle units

degrees are totally fine to use

just be clear

[amber hollow]

i understand

[solid needle]

now i think the other trig identities

at least the basic ones

you should be armed with everything you need to practice and get comfortable

now that you understand the true definition of sin/cos

things like the special values on the unit circle

once you know your 30-60-90 ratios

if you can just find the values for one quadrant, you know them all

so if you think you can take it from there, i will leave you with just two last tidbits

1. what is the significance of the tan function

2. why are csc and sec named that way

that okay with you?

[amber hollow]

yes

[solid needle]

ok let's start with tan function

tan = opp / adj

can you see that that is basically just y/x here

[amber hollow]

yea

[solid needle]

what does y/x remind you of

[amber hollow]

1?

[solid needle]

isn't y/x the slope of the blue line?

[amber hollow]

oh

yeah

[solid needle]

so this is the magic of tan

remember, trig function take an angle measure

they spit out a side length ratio

in the case of tan, it spits out the slope

[amber hollow]

oh

[solid needle]

so tan is a function that converts an angle direction into its slope

[amber hollow]

okay slope as in rise over run okay i see

[solid needle]

the inverse of this function does the reverse: you give it a slope, it gives you the angle measure

so maybe you might be able to understand now why the tan function is pretty important

[amber hollow]

it gives u the rate of change of something given an angle as long as you can make a right triangle with that angle?

[solid needle]

uh sorta?

you're splitting contexts now

so like you can make a right triangle sure

but then the rate of change of the hypotenuse

what does that mean?

just think about it on the coordinate plane

using our definition of what the angle represents

the new one

the angle t is less a measure of angle as it is a label for the direction from the origin

so if i say 180 deg it really means left

if i say 270 deg it really means down

[amber hollow]

is it the distance between 2 points on that plane?

[solid needle]

what is?

[amber hollow]

oh wait i was only thinking about the hypotenuse

uh

[solid needle]

think of it as a compass

select a direction, and then there is a t value that it corresponds to

imagine you're playing fortnite and i say "enemy team on your left, but slightly in front"

that's not as clear, but if I said "enemy team at 20 degrees"

you'd know EXACTLY where to turn to to see the enemy

on this unit circle, t is a direction, not an angle measure

[amber hollow]

okay i can understand that

[solid needle]

so on the coordinate plane

tan converts that angular direction to a slope value

[amber hollow]

so i understand slope refers to steepness and direction. but i mostly get that in terms of a graph. im not sure how it applies to the unit circle then

[solid needle]

the slope of the ray

that you draw from the origin

[amber hollow]

right

[solid needle]

so take this for example

suppose the angle here is 45 degrees

what's the slope of this ray

[amber hollow]

do i need to get the actual numbers or am i supposed to infer something

[solid needle]

you should know for a 45 degree angle

isn't it "midway" between 0 and 90?

[amber hollow]

yes

[solid needle]

draw a line with slope 1

how does it compare to that line?

[amber hollow]

isn't that a line with a slope of 1 already

[solid needle]

well yeah

that's what im saying

[amber hollow]

oh

[solid needle]

this line is a slope of 1

and also a direction of 45 degrees

and so

tan (45 deg) = 1

it converts direction to slope

[amber hollow]

i see

im just not sure why its important

[solid needle]

we often describe locations of things on a map using coordinates

but if you need to go from one place to another

you're usually not moving along straight gridlines

you're moving in a straight line from one point to another

so if you have two coordinates, a start and a destination

you can use the tan function

to calculate what direction you need the compass pointing in

[amber hollow]

you said angle is direction but tan takes the direction and turns it into a slope ?

[solid needle]

the angle and slope are both ways to describe direction, aren't they?

take any line, oriented in any direction

we can describe where the line is pointing by either specifying its slope or specifying the angle

it's kind of like converting units, like from inches to cm

we are converting two descriptors of direction, from angle to slope

but unlike with inches-cm, this conversion is non-linear, so it's quite tricky

the tan function describes this conversion in one function

[amber hollow]

oh okay

i think i kinda get it now

[solid needle]

ready for the last one? the naming convention of csc and sec

[amber hollow]

yeah go ahead

[solid needle]

ok so we had the pythagorean trig identity

sin^2 + cos^2 = 1

what happens if we divide both sides by sin^2?

[amber hollow]

cos^2 = 1/sin^2

[solid needle]

uh not quite

you need to divide both sides of the equation

[amber hollow]

whoops

cos^2/sin^2 = 1/sin^2

[solid needle]

uh

[amber hollow]

[solid needle]

(sin^2 + cos^2) / sin^2 = 1 / sin^2

does that help?

[dark sparrow]

what is the problem at hand anyway

[solid needle]

im just showing that if you divide both sides by sin^2

that's how you get the naming conventions of csc and sec

[limber ibex]

tan45

[dark sparrow]

no but like what problem(s) are you discussing rn

[amber hollow]

okay wait can't you just do sin^2/sin^2 + cos^2/sin^2 or am i tripping

[solid needle]

nothing, just a general trig overview

[dark sparrow]

right ok

[solid needle]

you can

that's right

[amber hollow]

okay that's where i got this from

[solid needle]

that's not the same thing though

[amber hollow]

oh

1

[solid needle]

start here and simplify one step at a time

[amber hollow]

i forgot

[solid needle]

yeah

[amber hollow]

im going to sleep after this

[limber ibex]

trigonometry is easy

[solid needle]

np

i threw a lot at you

if you want, we can call it here

this last bit isn't that important

[limber ibex]

but many identities and formulas are there

[solid needle]

take some time to digest it

[amber hollow]

no i wanna hear it it's all right

[limber ibex]

therefore derivation is important

[amber hollow]

okay so 1 + cos^2 = 1/sin^2 i understand that

[solid needle]

close

cos^2/sin^2

[limber ibex]

in every proof or formulae

[solid needle]

you forgot the /sin^2

[amber hollow]

okay pretend i didnt carry on

[solid needle]

so now let's use our other trig definitions

can you see this gives:

1 + cot^2 = csc^2

[amber hollow]

yeah

[solid needle]

so this is why

1/sin = csc

1/cos = sec

why they're swapped

because it's not the reciprocal identities that are the most important

it is the pythagorean identities that are more important

if you had divided both sides by cos^2 instead of sin^2

you would get

tan^2 + 1 = sec^2

[amber hollow]

tan is just the inverse of cos right

[solid needle]

nope

[amber hollow]

cot*

[solid needle]

yes

er

reciprocal technically

or multiplicative inverse

but yes

[amber hollow]

all right

i'll sleep on this all and digest what you've taught me today

[solid needle]

cool

hopefully trig doesn't feel as obtuse anymore

[amber hollow]

i understand trig way more than i ever did now yeah

tysm much internet stranger

honestly i appreciate it

[solid needle]

np

heading to bed now

good luck

[upper karma]

Should i learn sum and difference of trig functions formulas

I mean, sin a + sin b

There are lots of formulas in trig and i dont know whether i should remember this too

I mean is this formula common and often used?

[dark sparrow]

somewhat

sum->product and product->sum are both in use

the latter is needed for some integrals for example

[upper karma]

Ok

Also, if my school skipped teaching trig inequalities and recently we've begun learning calculus, should i learn these inequalities on my own

Are they needed for basic calculus

[dark sparrow]

wouldn't hurt

[solid needle]

they arent used super often, depending on how far you go in math and what you study, but when you do need them, nothing else will save you

you should learn them

also, i find the derivation of the sum formulas is more important than just knowing the formula by memory

if you have the time to investigate, you should read up on complex numbers and polar form, then use it to derive the sum formulas

it will not only make the formulas waaaay less annoying to memorize, but it really adds a lot of cool and important insight

[devout berry]

Im stuck on a geometry exercise for my math assignment, can some1 help me

stupid math server i never get help

[sturdy stone]

You only waited 3 minutes

And you also didn't even post the problem

[trail tendon]

☠️

[summer oak]

Can someone please help me in geometry i have a quiz tomorrow and im in chapter 11 and im lost.

[trail tendon]

you'd probably want to take the area of the circle and subtract by the area of the pentagon, right?

[summer oak]

yes, but you need to find the area of the triangle first

i just dont know what to do

[trail tendon]

you mean if you divide the pentagon into triangles?

or wut

[summer oak]

yes

[trail tendon]

i mean

you probably need to find the area of the circle, and the area of the pentagon so that you can subtract

i think it would be better if you found the radius

because the radius will help you with a lot of it I think 💀

[summer oak]

i dont know how to..

[trail tendon]

did you draw the triangles in the pentagon?

[summer oak]

yes sir

[trail tendon]

what are the angles of the triangles?

oh i should probably say

two of the sides of each triangle are the radius of the circle

right

[summer oak]

yes

[trail tendon]

so can you find the angles of each triangle?

(if you find the angles, you can use trig to figure out the rest of the sides)

[hidden bone]

nr^2/2(sin2pie/n)

it is a formula for any shape inside a circle

[trail tendon]

really?

what is n btw

[trail tendon]

you mean any shape inscribed in a circle?

[hidden bone]

number of sides

[hidden bone]

in the sense like pentagon hexagon

like that

[trail tendon]

ah

but yeah i think it would have to be inscribed in the circle

[hidden bone]

question is the same right?

[trail tendon]

yeah, inscribed just kinda means its the largest shape while still being completely inside basically

so the farthest out points will touch the circle but not cross it

cuz it wouldn't make sense if it was outside the circle 💀

[hidden bone]

no this is area of pentagon formula

ohhh i get it

[steel anvil]

lol

If yall need help I’ll help

I already started

[cerulean turtle]

How do I answer this

Help

[steel anvil]

m<1 = m<3. The value of x will work no matter what as long as that’s true

||2(x+5)=(x+40)
2x+10=x+40
x = 30||

[steel anvil]

Hopefully understandable

And readable

[cerulean turtle]

subtract the 40 and 10?

How did it become 30?

[dark sparrow]

!nosols @steel anvil

[lime crown]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[wary jolt]

?

[dark sparrow]

@wary jolt any progress on this?

[wary jolt]

np

no

[dark sparrow]

ok

[wary jolt]

i only have one more try

[dark sparrow]

wait, is this a test?

[wary jolt]

no

[dark sparrow]

oh it says "Attempt 2 out of 2"

[wary jolt]

assistment

[dark sparrow]

assistment?

ok whatever

[wary jolt]

Assignments

[dark sparrow]

do you see the shaded stuff on your picture

y/n

[wary jolt]

@dark sparrow i skip everyday

[dark sparrow]

you skipped literally every single one of your math classes?

[wary jolt]

302 abences

[dark sparrow]

so you are bragging about never, or almost never, having gone to any math class?

or what

[wary jolt]

@dark sparrow help

[dark sparrow]

no

[dusky locust]

rosted

[obsidian harness]

You should really consider learning a trade if you're not that interested in school

Cause college is going to be more classes and you won't get anything out of that education if it's not for you

https://www.youtube.com/watch?v=4p5286T_kn0

There was this kid who got sent to the principal's office for drumming on the table in class

The principal encouraged him to take up drumming

He's gone real far now

[inner vapor]

47 absences to piano class

I was literally looking for a piano teacher earlier this week

Same with Physical Education class, people don’t attend it and hate it on hs but end up paying for gym memberships later on

Kinda bizarre

[bronze wind]

Can someone give me a rundown on how I would solve #1 so I can know how to do #2?

(I suck at trig so please dont call me stupid thanks)

[dark sparrow]

do you know the SOH-CAH-TOA mnemonic?

[bronze wind]

yes

[dark sparrow]

right

so for #1, let's say we want sin(R). which part of SOH-CAH-TOA will we need?

[bronze wind]

opp/hyp

@dark sparrow ^^

[dark sparrow]

right,

so then,
what is the length of the hypotenuse in our case?

[random urchin]

I need help with questions 1-6 I don’t understand this topic at all

[dark sparrow]

@bronze wind

[bronze wind]

34

wait

is my answer 8/17

for SinR

[dark sparrow]

i'd rather you didn't jump ahead

but yes it is

[bronze wind]

am I able to flip it for SinS

[dark sparrow]

sin(R), not SinR

[dark sparrow]

flip?

[bronze wind]

yeah like 17/8

[dark sparrow]

no

[bronze wind]

ok

[dark sparrow]

sin(S) = (opposite side to S)/(hypotenuse)

[bronze wind]

ok

thanks

[paper epoch]

Hey can I get some help

[dark sparrow]

step 0 of any geometry problem: make a diagram.

have you made a diagram?

[upper karma]

Looking for some math resources. (Ping if posible).

[north kindle]

How would I find the area of the shaded region?

Better pic

I’m not familiar with finding areas involving parabolic curves like these

!status

[lime crown]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[north kindle]

Currently 1

[lapis junco]

Have you been helped?

[north kindle]

Yeah

I was being dumb and forgot to square something

Made it much simpler

[lapis junco]

Okay bc I would say take an integral

At least it’s one way

[north kindle]

Yeah I don’t know calc lol

[lapis junco]

Oh duh that’s why you sent it here lmao I’m so slow

[north kindle]

What does “the arc between P and Q” mean exactly?

[cunning lion]

an arc is a segment of a circle

[regal phoenix]

solve what

[steel anvil]

Meow

[bright harbor]

im sure its the edge of the circle, not the whole 2d area part right?

[steel anvil]

Wassup

[bright harbor]

wassdown

[steel anvil]

Points on the circle. Then it’s asking for the distance between these two points.

[steel anvil]

Money

[steel anvil]

So what is the probability that it ends up being less than or at 60 degrees

[bright harbor]

60/360 = 1/6?

[steel anvil]

I could be reading to literally but it’s just 1/3

If it’s 300 degrees, it’s also 60 degrees

I’d assume 1/3

[bright harbor]

i think its 1/6

[steel anvil]

1/6 probably best to put

But I’m sure you can see why I also think 1/3rd would make sense? Think of being 330 degrees. You’re basically 30 degrees

[bright harbor]

why would it be 1/3 prob.?

[steel anvil]

I’m reading too far into it

Just put 1/6

[north kindle]

It was 1/3, as I thought

[steel anvil]

1/3?

I knew it’

Ughhhhh

I hate reading too far deep. Into shit

It makes me question myself.

[hidden tartan]

@jolly vapor Wanna chat in here?

Is there anything specific you're confused about? If not, can you tell me what you've been covering recently?

[jolly vapor]

yess thank you @hidden tartan

[hidden tartan]

No prob!

[jolly vapor]

try not to laugh but this is what im doing

[hidden tartan]

Okay, and you're having trouble with all of these?

If so we can start at 1, do you know how to get the circumference of a circle?

Or is there a specific concept you wanna talk about

[jolly vapor]

u need to get the diameter first

radius BY 2

[hidden tartan]

Yup, that or the radius

Yeah

[jolly vapor]

and then multiply it by pi

so lets see

[hidden tartan]

You don't need to fully expand it btw, you can often leave pi symbols in your answer

[jolly vapor]

can u tell me how i would do 1.

and then i can try 2 and 3 and 4

[hidden tartan]

I don't know how your class wants it submitted, but in your scratch work there's nothing wrong with leaving 2π as 2π

[jolly vapor]

yes

[hidden tartan]

Okay so, first in 1 I'd get the circumference of the circle, which should be 24π yeah?

[jolly vapor]

yeah because 12 x 2 is 24

that would be the radius

right

because its from the middle to one side of a circle

[hidden tartan]

👍 Now, we want to restrict it to just the length between A and B

Since the angle between A and B is 90 degrees, that means the arc between them is one quarter of the circle

So we would want 1/4 of 24π

Can you simplify that for me?

Feel free to let me know if you're stuck or confused

[jolly vapor]

making dinner 1 sec

hi

im back

[hidden tartan]

Yo

[jolly vapor]

so ok

so restrict it wdym

[jolly vapor]

why do u say that

because 90 x 4 is 360?

[jolly vapor]

i can use desmos right

or wouldnt that just be

1/4 of 24

[hidden tartan]

Hmmm... I mean we want to measure specifically the space between A and B

[jolly vapor]

so 6

[hidden tartan]

Yes

[jolly vapor]

6PI

[jolly vapor]

6π

[hidden tartan]

Yup!

[jolly vapor]

it says round to the nerest hundreth

nearest hunderth*

[hidden tartan]

You can use a calculator for that

[jolly vapor]

hundreth

what

hundereth

hundredth

[hidden tartan]

Yeah, two decimal places

[jolly vapor]

nearest hundreth would be

18.84

[hidden tartan]

18.85 actually, the 9 rounds up

[jolly vapor]

yes

[hidden tartan]

So does that answer make sense? If not I can walk you through some intuition

[jolly vapor]

yes kind of

the most annoying problem with me is

i learn one concept

and then theres a different question with a slightly different concept

and then im brand new, all confused again thats why i dont ask for help from my teacher

i used to be really smart in elementary top of my game

[hidden tartan]

I think that's normal, one of the major things you want to get out of learning math is getting better at abstracting and reapplying concepts

Anyway, wanna move onto 2?

[jolly vapor]

yes please

thanks for helping me

[hidden tartan]

No problem 🙂

[jolly vapor]

do you have cashapp

[hidden tartan]

Nah, don't worry about that

[jolly vapor]

also

should i go grab a sheet of paper

so all this stuff doesnt fly through the back of my ear

cause you know what they say, when you write stuff down its easier to memorize then just using your eyes

[hidden tartan]

it couldn't hurt ¯_(ツ)_/¯

[jolly vapor]

alright

[hidden tartan]

I like doing my work on paper

[jolly vapor]

it isnt really a preference thing, ive heard youll fall miserable in math if you dont write stuff down

[hidden tartan]

Oh I just double checked, the question says to use 3.14 for π, and 6π = 18.84

[jolly vapor]

is that wrong

[hidden tartan]

Sorry, what I meant is that earlier I said it was 18.85 since 18.849... would round up, but that's only if you use the longer definition for π

But anyway, lets get to question 2 alright?

So the steps for solving this are similar to 1

You will need to get the area of the circle, and find out how much of the area you actually want

[jolly vapor]

so the radius is given to us

right

so you would do

pie multiplied by 12 to the power of 2

so 3.14 x 12 to the power of 2

452.16

then it says 60 degrees

[jolly vapor]

so wouldnt u divided this by 6

452 divided by 6

[hidden tartan]

Okay cool, I was a little worried you mean (π12)^2, glad you actually meant π(12^2)

Yup!

[jolly vapor]

am i yapping

or did i do it right

[hidden tartan]

Yeah you're right I believe, can you give me the answer you got to?

[jolly vapor]

um some long decimal but

let me double check

give me 30 seconds

[hidden tartan]

I think you might be simplifying too soon

Instead of 452/6, maybe do 452.16/6

[jolly vapor]

im allowed to use calculators on test dont worry lol

so that answer would be 75.36 right

[hidden tartan]

Yup!

[jolly vapor]

woohoo

so right now we're at

18.84+75.36

[hidden tartan]

Nothing wrong with using calculators for stuff like this, though for good practice you should always try doing some stuff by hand

Great, now lets move onto question 3

[jolly vapor]

especially going into algebra II

[jolly vapor]

alright

[jolly vapor]

similar concept

so they gave us the radius once again

[hidden tartan]

Yup! Do you know how much of a circle 120 degrees is?

[jolly vapor]

1/3

yeah

1/3

[hidden tartan]

👍

[jolly vapor]

so basically

pi multipled by 6 to the power of 2

[hidden tartan]

Oh not quite, we're finding arc length not area

[jolly vapor]

f

ok so

[hidden tartan]

lmao it's all good, mistakes are part of the process

[jolly vapor]

let me just use this for refrence

https://media.discordapp.net/attachments/326138757474680852/1229947864109809706/image.png?ex=66318936&is=661f1436&hm=84225ac8f8fce5287a6b7618788d15fccd2a9aa92850a311b22e59d9cfc25cdb&=&format=webp&quality=lossless

so multiply 6 by 3????

idrk

[hidden tartan]

Okay, well lets start with getting the circumference of the whole circle

[jolly vapor]

yeah

so the radius is 6

if we multiply 6 by 2, we'll get 12

which means d = 12

12 multipled by 3.14

37.68

[jolly vapor]

circumfrence

[hidden tartan]

Yup!

[jolly vapor]

wait

dont say anything

wait

wait

let me cook

wait

[hidden tartan]

Side note, I mentioned earlier you can leave things like that as 12π until the end. You don't have to do that if you're not comfortable with it, but it's just a small simplifying measure

I'll let you cook 🙂

[jolly vapor]

what do you mean

[hidden tartan]

Just, instead of working with 37.68, you could work with 12π

We can talk about this later, keep cooking

[jolly vapor]

so we know the radius and circumfrence

so i would have to

wouldnt that be the side length

120 would be the central angle

wait, is center angle and degrees of the arc the same thing

[hidden tartan]

I think so, though I'm not sure what notation your class is using

By side length do you mean arc length?

[jolly vapor]

^^^ yes

[hidden tartan]

Then yes!

Using 3.14 instead of π gives 12.56, so you should use that in your answer

[jolly vapor]

18.84+75.36+12.56

[jolly vapor]

Okaay

now these two are practically same thing

so if i dont get it, its a matter of short term memory loss not understanding quickly

[hidden tartan]

Very similar yup!

[jolly vapor]

so i would have to

pi multipled by 9 to the power of 2

right

[hidden tartan]

Yup! π9^2

[jolly vapor]

and also

does diamaater always have to be in a straight line

like a straight 180 degree angle, ALWAYS

it cant curve like this or else that would be radius correct?

[hidden tartan]

Eh.... kinda? All the diameter is is the distance from opposite sides on the circle

[jolly vapor]

found my answer btw

[hidden tartan]

The radius is the distance from the center of the circle to the edge

[jolly vapor]

Does diameter have to be a straight line though

Like does diameter always have to be straight

[hidden tartan]

Well diameter doesn't really mean line

[hidden tartan]

It's more like a straight line going from one side to the other and passing through the center will have the length of the diameter

[jolly vapor]

so it could be curved like that right

[jolly vapor]

exectly

exactly*

[jolly vapor]

okay back to the thing

https://media.discordapp.net/attachments/326138757474680852/1229950595981639771/image.png?ex=66318bc1&is=661f16c1&hm=9913f911dd9b30a88c844b9176e2ceff745db558567c057e2d236b07f8b050c4&=&format=webp&quality=lossless

254.46 would be the what again

[hidden tartan]

In a sense though, since d = 2r, any line that going straight from the edge to the center then straight to the edge again would have the length of the diameter

That would be the area of the whole circle

[jolly vapor]

alright

so...

would i do

254.46/360

[hidden tartan]

Hmm, well that could be one way of getting there but it depends on what you want to do with that value

[hidden tartan]

So all we want is the area covered by 160 degrees right?

[jolly vapor]

yes

[hidden tartan]

So like a pie slice with angle 160

[hidden tartan]

Yeah, that's like 4/9 or something

[jolly vapor]

yes it is

so i would do

4/9 of 9??

idk

im missing one step lol

[hidden tartan]

Okay, so the shaded region makes up 4/9 of the whole circle right?

[jolly vapor]

yea

[hidden tartan]

And the whole circle has area 254.34

[jolly vapor]

yeds

so 4/9 of 254.34?

[hidden tartan]

Yup!

[jolly vapor]

right

[hidden tartan]

Looks right to me 🙂

[jolly vapor]

so look

instead of just letting it sit in my head

how about we making a formula i could write in my notebook

[hidden tartan]

Sure!

Oh, I just noticed that question 4 says to round to the nearest whole number

[jolly vapor]

113

[hidden tartan]

👍

[jolly vapor]

we got it right

but the thing is theres kids in my class

[hidden tartan]

Awesome!

[jolly vapor]

that sped through this in like 5 minutes

i want to get on their level lol, it should come natural to me, dont know where i fell off

[hidden tartan]

Well I don't know how to help with that unfortunately, but if you're feeling left behind you definitely could talk to your teacher, especially about the stuff you missed earlier on

But also here's the thing

You're in grade 8 right? You've got TONS of time to figure things out

[jolly vapor]

true

Next year im taking AP Human geography

[hidden tartan]

Sounds nuts

[jolly vapor]

this question basically is the same concept as this right

[hidden tartan]

Yeah I think so

[jolly vapor]

so first think i should do is

multiple 7 to 3.14

then to the power of 2

[hidden tartan]

Hmm, no you only take it to the power of 2 if you want the area

Remember:
Circumference = 2πr
Area = πr^2

And an arc is a line

[jolly vapor]

okay so

circumfrence ^

[hidden tartan]

Looks good so far, now you just want to take the part of that that 125 degrees covers

[jolly vapor]

easy

that sounds right

to me

[hidden tartan]

👍

[jolly vapor]

round to the nearest tenth

so 15.3 right

[hidden tartan]

Yup

[jolly vapor]

beecause 6 is above 5

alr

15.3 + x + x + x

LOL

that was soo funny

[hidden tartan]

Hmm, I think I'm gonna have to go pretty quick

[jolly vapor]

your good, i dont wanna bother u havea goodnight

ill text you tommorow because i have a big summative test and let you know my score

[hidden tartan]

I need to get a little more work done before my meds wear off

Sounds great! Good luck 🙂

[finite basin]

How necessary is geometry to master for someone who isn’t going to be a math major?

I have the opportunity to take geometry over the summer and be in algebra 2 next year and I’m not sure

I’m a good student overall but would it be better to learn geometry step by step rather than being rushed ?

[autumn pelican]

I am overthinking and confusing myself.

I am given one of two non-right angles, 67.5, and the hypotenuse 425, so from using Pythagorean Theorem I concluded my adjacent equals ~ 419.61
So if I am asked to find side lengths and angles inside, is it from here I just plug into trig functions to get the angles? I'm at the end of the day and can barely think.

[trail tendon]

do you know if the triangles are right triangles? and are you saying one side length is 67.5 and the hypotenuse is 425?

[autumn pelican]

The wording I was given is new to me so I am probably making mistakes. Yes it is a right triangle and the vocabulary used is

"one of the two non-right angles is 3pi/8 and the hypotenuse is 425 units long"

[trail tendon]

is that the entire problem

or does it say before that theres a right angle

[autumn pelican]

Heres the entire problem

Solve for all the side lengths as well as the angles inside of a right triangle if one of
the two non-right angles is 3𝜋/8 and the hypotenuse is 425 units long.

[trail tendon]

ahh

[autumn pelican]

I don't neccessarily want the answer, I am just trying to figure out what I am wrong on

[trail tendon]

so you have two angles and one side length (the hypotenuse) right?

[autumn pelican]

That's what I believe yes

[trail tendon]

wait sorry

two angles

mb

[autumn pelican]

And I concluded 3pi/8 is my opposite

[trail tendon]

um

opposite to what?

[autumn pelican]

adjacent?

[trail tendon]

i mean...

usually you talk about the sides with reference to the angle that you have

so you would call the side next to your angle adjacent, and the one farthest away from your angle the opposite

and then the longest side is always the hypotenuse

right

or does that not make sense

[autumn pelican]

I suppose it does, I just haven't been given many references to get this experience

[trail tendon]

ahh

anyway

[autumn pelican]

I always thought, opposite is the shorter leg, adjacent is the 2nd longest, and obv hypot is the longest

[trail tendon]

ahhh nah

they just mean opposite the angle or adjacent to the angle

so those can actually be swapped depending on the angle you're talking about

anyway

[autumn pelican]

The angle being the hypotenuse?

[trail tendon]

no, the hypotenuse is a side

let me get a picture one sec

[autumn pelican]

That'd help

Do you mean the box placed that references it as a right triangle?

[trail tendon]

so the hypotenuse is always just the longest side

[autumn pelican]

mhm

[trail tendon]

i pikced the blue angle

so the opposite would be the length opposite the angle

the adjacent would be right next to the angle

right?

but

[autumn pelican]

So.. by angle you mean whcih way it is facing?

[trail tendon]

oh

do you see the blue curve thing

[autumn pelican]

Yes

[trail tendon]

thats the angle

right

[autumn pelican]

Right

Isn't that always where the angle is though?

[trail tendon]

nah look

what if it were that angle

now the side length you call "opposite" is different

right

[autumn pelican]

I see I see

[trail tendon]

it's all with reference to the angle

[autumn pelican]

IN my case, I wasn't given a reference image but only this prompt, so I cna't really yassign an opposite yet can I?

[trail tendon]

besides hypotenuse is always the longest lol

[trail tendon]

we can draw a picture

right?

[autumn pelican]

uhh

I suppose, but how do we assign the angle?

[trail tendon]

we know one angle is 3pi/8 or smthn right

[autumn pelican]

Yeah, and the hypotenuse is 425, I think

[trail tendon]

ok

[autumn pelican]

3pi/8 is 67.5 degrees (unless we aren't converting to degrees)

[trail tendon]

there's no need unless you really want to...

its usually better just to work with radians

but degrees is fine too, which one do u wanna work with

[autumn pelican]

It depends what I am doing with them, either work for me though

[trail tendon]

alr ill just keep it in radians

the triangle is going to look something like this, yeah?

[autumn pelican]

mhm

[trail tendon]

i could have put the angle on the opposite corner, the diagram isn't accurate anyway

so u know the opposite and adjacent according to this angle right

[autumn pelican]

Yes

[trail tendon]

alr

im just gonna write it in

and by the way

can you find the last angle?

in the corner?

just from here

[autumn pelican]

My instinct was to use pythagorean theorem but it didn't seem r ight

[trail tendon]

uh

[autumn pelican]

I am not completely accustomed to the triangles relating to trig functions

[trail tendon]

ahh ok

so the reason you can't use pythagorean theorem

is because

lets just say
x = opposite
y = adjacent
sqrt( x^2 + y^2) = 425, right?

[autumn pelican]

yes

[trail tendon]

you have two variables and one equation

you cant solve an equation with two unknowns unless you have two equations or smthn

right

[autumn pelican]

Oh, I am further confusing m yself. I thought x = 3pi/8

[trail tendon]

ohh nah

3pi/8 is the angle

these would be side lengths

opposite and adjacent are side lengths

right

[autumn pelican]

I see, yeah

[trail tendon]

yeah

the pythagorean theorem is used on right triangles when you have two side lengths and you want to find the third

here you only have one

but you have two (technically 3) angles

speaking of which, can you find the third angle?

[autumn pelican]

Wait, I thought the hypotenuse was a length

[trail tendon]

it is

[autumn pelican]

I assume I am meant to use a trig function, but since I don't havean opposite or adjacent I don't know how

[trail tendon]

ah

really quick tho do you know how to find the third angle

the blue one here

[autumn pelican]

I don't

[trail tendon]

ok

what if you put everything in degrees

3pi/8 = 67.5 degrees, and you have a right triangle which is 90 degrees

do you know how to find the last angle

[autumn pelican]

90 - 67.5

?

[trail tendon]

uh not exactly

do you know what the angles of the triangle have to add up to?

[autumn pelican]

180 - 90 - 67.5?

[trail tendon]

yes

so whats the last angle

[autumn pelican]

22.5

[trail tendon]

yeah

so now we have to do the trig

to find the two side lengths that we don't know

[autumn pelican]

So all our angles isnide the triangle are 67.5, 90, and 22.5?

[trail tendon]

yep!

[autumn pelican]

This is probably stupid, but in which areas do they reside in the triangle? or does it matter where I visually place them

Ok, well 90 makes sense at least lol

[trail tendon]

XD 1 sec

[trail tendon]

basically you know how theres 2 more spots

[autumn pelican]

yes

[trail tendon]

it could be in either one, more or less

[autumn pelican]

Ah, so doesn't really matter?

[trail tendon]

because the diagram isn't to scale anyway

[autumn pelican]

oh ok

[trail tendon]

yeah

like you can swap the blue and red angles it doesnt really matter

[autumn pelican]

So are these angles respectively capital A, B and C?

and lower cases are length?

[trail tendon]

uh

i don't know

[autumn pelican]

Probably a teacher dependent thing

[trail tendon]

yeah lol

i mean i think there might be a common choice

but i dont remember lol

[autumn pelican]

lol all good

[trail tendon]

i think that would be the common choice though probably

[autumn pelican]

I just looked at my diagram, c is hypotenuse so I am going to assume it is

[trail tendon]

alr XD

[autumn pelican]

lol

[trail tendon]

so uh

do you know the three main trig functions

[autumn pelican]

Yes

[trail tendon]

what are they

[autumn pelican]

Sine, Cosine and Tangent

[trail tendon]

yea

do you know what they equal

like in terms of sides

[autumn pelican]

Sine = opposite/hypot, Cosine = Adjacent/hypot, Tan = opposite/adjacent

[trail tendon]

yes

do you know the opposite?

[autumn pelican]

No

[trail tendon]

by the way, lets just use the reference angle 3pi/8

if thats ok

[autumn pelican]

It is

[trail tendon]

i mean we could have used 22.5 or whatever if we wanted, but yeah

it would have swapped the adjacent and opposite and we woulda gotten same answer

[autumn pelican]

Same thing, I'll prob just mentally convert it lol

[trail tendon]

XD

naw i meant like 22.5 as in the other angle

besides 67.5 (3pi/8)

but yea

do we know the adjacent side?

[autumn pelican]

No, don't we only know the hypotenuse?

[trail tendon]

yes

sin(x) = opposite/hypotenuse, right?

[autumn pelican]

yes

[trail tendon]

we know x, which is the angle, and we know the hypotenuse

do you know how to solve this for the opposite?

by the way you can plug sin(some angle) into a calculator to get a value

[autumn pelican]

Do we do algebra now? I haven't done this yet or I don't remember

[trail tendon]

um

it's more like trigonometry/algebra kinda yeah

[autumn pelican]

alr, so would we multiply the hypotenuse then?

[trail tendon]

wdym

by both sides? or wut

[autumn pelican]

Yes

multiply hypotenuse on both sides to cancel out one side

[trail tendon]

yes

[autumn pelican]

sin(x) * hypot = opp

[trail tendon]

yep

so whats x

[autumn pelican]

67.5 or 22.5?

[trail tendon]

we're using 67.5

[autumn pelican]

ok

[trail tendon]

so yeah

and then you can plug in the hypotenuse

u prob have calculator right

[autumn pelican]

yeah

[trail tendon]

make sure its in degrees if ur using degrees

[autumn pelican]

one sec let me find it

I feel like I did sometghing wrong, opposite = 9562.5

[trail tendon]

uhhhhhh

[autumn pelican]

oh, I forgot sine

[trail tendon]

oh LOL

[autumn pelican]

lol

[trail tendon]

yeah that might make it a bit off 😂

[autumn pelican]

😂