Roots of trigonometric functions...Help pls

[icy jacinth]

Hello, okay so the deal here...
I need to express the roots of the following trigonometric function, in a single set... That's what we got in class... The teacher told us that we only needed to determine the value of A and that there was a division
Could someone enlighten me please???

[ancient estuary]

@icy jacinth What are roots of a function?

[icy jacinth]

The intersections with the x-axis

[ancient estuary]

great !

[ancient estuary]

how do you find them then?

[icy jacinth]

by setting the function equal to 0 and solving for x… but since it is a trigonometric function, it has infinitely many roots for each period of the function

[ancient estuary]

you are good at this ha

when solving trig equations i always suggest

oops wasnt done yet

the phrasing at the end isnt entirely correct but i see you have the idea

so you set -2cosx + 1 = 0 and you get cosx = ?

[icy jacinth]

yeah I got 1/2 so x = pi/3

[ancient estuary]

nice cosx = 1/2

[ancient estuary]

however x can be many things as you have said

x can be pi/3 it can also be 7pi/3

it can be 13pi/3, -pi/3 ...

there is something x can be between pi/3 and 7pi/3 would you know what that is and how to get it?

[icy jacinth]

I nderstood that part one root is π/3… the pattern is π/3, 5π/3, 7π/3, 11π/3… forming a sequence of 4π/3, then 2π/3, then 4π/3, and so on, repeating like this. What my teacher wants is for us to express this in a single set. The problem is that he wants it without using ±, and that’s it… my brain can’t handle any more. He wants a rule that gives any root of that function, one for eveything

[ancient estuary]

how did you get 5pi/3?

[icy jacinth]

Is it half of the period, right? for the first root I was supposed to subtract the period of the function… although honestly, I just iterated values that gave me 0 and that't it

[ancient estuary]

i encourage to always have a sure understanding of everything in maths

maths is like a house ... if the foundation is bad the house will collapse

did your teacher teach you how cos and sine relates to the unit circle?

[icy jacinth]

yeah

[ancient estuary]

nice

pardon my long winded answer but i believe in teaching how to fish than giving fish

how does cos relate to the unit circle?

its ok to not know ... whenever stuck in maths ... id also encourage going as deep as possible to try to find where the problems lie... why i am asking many questions

then we can help from there

[icy jacinth]

Okay, don’t worry… honestly, I really appreciate it, you’re making me think haha okay

[icy jacinth]

okay so.... If I take a bit too long its cuz Im not an english speaker so Im like trying to figure out how to explain it hahah okay.. soo

well, from the unit circle perspective, since we are working with the trigonometric function and not the trigonometric ratio, we know that cosine is the division of the x-coordinate of the point by the radius of the circle... or in other words, the coordinate on the x-axis divided by the radius, considering that the angle must be a real number in radians, cuz degrees are not real numbers in this context, and so on... or, more simply, using a right triangle: cos is the adjacent side over the hypotenus

[ancient estuary]

great! i see you have the idea.

so if you have a unit circle ( circle with radius 1) and draw any point on it ... cos will tell you how big the x coordinate is

give me a sec

[ancient estuary]

so if you have an angle x it will take you to a point on the circle

and cosx tells you how much to the right or left the point is sinx tells you how much up or down you will have to get to that point

remembering that the radius of the circle is 1

[icy jacinth]

okay, yeah 🗒️ 🐊

[ancient estuary]

so if x is 0 you see that

[ancient estuary]

you need to go 1 unit to the right to get that point

that means cos0 is 1

you also see that you dont need to go up at all that means sin0 is 0

and you can do this for all angles and this is where cos and sin comes from

remember also if you go 360 degrees round a circle you come to the same point

[icy jacinth]

okay, right 📝🐢

[ancient estuary]

do you know pythagoras?

and dw we will soon get to our answers ha

but this will be the only time you will need to learn all this...

if x is 60

as cos60 is 1/2 it means when x is 60 we need to go 1/2 to the right to get to the point

and as sin60 is (root3 )/2 it means that is how much up we need to go

[icy jacinth]

Yesss

[icy jacinth]

Oh, okay I see

[ancient estuary]

sorry my laptop just bugged out but you also see why adding 360 or 2pi to an angle gives the same answer for cos and sine as you get to the same point on the circle

[ancient estuary]

do you understand that adding 360 / 2pi to an angle rotates around the circle?

[icy jacinth]

Yup

[ancient estuary]

you then understand why adding 2pi then gives the same answer for cos and sin?

[icy jacinth]

I think I do

[ancient estuary]

hmmm.... good

make sure everything is crystal clear

[ancient estuary]

what isnt perfectly clear?

[icy jacinth]

Hahaha sorry I do understand, I'm just afraid of the next question 😭😭😭

[ancient estuary]

sure?

[icy jacinth]

Sorry I feel like I can say that I understand it and then you're going to ask me what that has to do with the color of the sun and then I don't know anything 😭😭😭😭
You are really good explaining, sorry
But yes, I understood, there's no problem

[ancient estuary]

alright we will move faster now

a few things to note if the point is on the left cosx will give a negative answer... and if the point is down sinx will give a negative answer

[ancient estuary]

because the point we get when x is 120 is on the left we get a negative answer for cos120
but we are still going up to get to the point so sin120 will give a positive answer

you can type cos120 in your calculator to check and sin120

but also we are very smart and can now see something
do you know why the angle 60 is there on that diagram?

[icy jacinth]

Yup

[ancient estuary]

so why is 60 on that diagram?

[icy jacinth]

😭😭😭
hahaha I'm scared of saying something stupid 😭😭😭

Okay

[ancient estuary]

a wise man once said never be afraid to fail

only then can you learn 🙂

[icy jacinth]

As far as I understand, okay first, okay...
Ah, I don't know how to explain it, okay so... Seeing it from the point of view of...
Well, it's basically 180-120, right?
We are taking the "real" angle... The sin is still in the positive quadrants and the cos has already moved to the second quadrant

[ancient estuary]

hmm...

ahh yes !! ha

its 180 - 120

because the sum of angles on a line is 180

now because we are very smart we can start to see patterns

[icy jacinth]

Hahaha

[ancient estuary]

see going an angle of 60 from the right got us to a point on the circle (x = 60) if we go an angle of fake 60 on the left we get to another point

[ancient estuary]

for the two points do we have to go up to the same amount to get them?

[icy jacinth]

Yes?

[ancient estuary]

absolutely

[ancient estuary]

this means that sin(60) and sin(120) are the same because you have to go up the same amount to get them

[ancient estuary]

does that make sense?

[icy jacinth]

Yes, yes

[ancient estuary]

however if you have to go the right some amount to get to point you will have to go the left of the same amount to get to another point

and remember if you go left you have a negative answer... so if i have cos(60)

cos(120) = - cos(60)

[ancient estuary]

does this make sense

[icy jacinth]

yup, yup

[ancient estuary]

are you absolutely sure?

[icy jacinth]

100%

[ancient estuary]

great then but just to be clear... if i need to go half to the right to get to my point ( which is given by cos(60)) i will need to go half to the left to get to my point when x is 120 but because im goint to the left cos120 will give me -1/2 im going halft to the left
so cos120 = -cos60

make sense?

[icy jacinth]

Makes sense

[ancient estuary]

great we can then generalize these pattern

as we can see that when we have an angle x and have an angle (180-x) you go up the same amount we can say sin(x) = sin(180-x)

[ancient estuary]

it is quite confusing. does it make sense ? and if it doesnt let me know

[icy jacinth]

Omg I think you just opened my eyes, okay,yeah we're good, I understand

[icy jacinth]

We're doing good :)

[ancient estuary]

are you absolutely sure

[icy jacinth]

110%

[ancient estuary]

and you can always check with your calculator too pick any number for x... say try 30

put sin30

then put sin150

sin(180-30) = sin(150)

try sin20 and sin160

and so on

[icy jacinth]

okay we're good

[ancient estuary]

what is the answer for sin5pi/6

[icy jacinth]

1/2

[ancient estuary]

then we can also say cos(x) = - cos(180-x)

confusing

[icy jacinth]

Oh no, I think I get it right

[ancient estuary]

what grade are you in by the way?
but you can see that for cos(x) we go some amount to the right but for cos(180-x) we go the same amount to the left but because we are going to the left cos(180-x) will give us the negative of cos(x)

so if you try cos(60) you will get 1/2

but cos120 = -1/2

try cos(30) then try cos(150

[icy jacinth]

I prefer not to reveal information that will make people make fun of me 😭😭😭😭

[ancient estuary]

why would someone make fun of you for your grade?

[icy jacinth]

I don't know 😭😭😭
||I'm in the first semester of engineering||😔😔😔

[ancient estuary]

university?

[icy jacinth]

Yeah

[ancient estuary]

i only learnt about how cos relates to the unit circle after i finished high school

sometimes our teachers arent the best... and we shouldnt judge someone by their knowledge at some level... it doesnt matter anyway just learn what you dont know ha

[icy jacinth]

Okay thank you 🥲

[ancient estuary]

i only asked so i dont go too deep

[ancient estuary]

does this make sense?

[icy jacinth]

yup

[ancient estuary]

we then get to the pattern that relates to your original question ha

[ancient estuary]

if we have an angle (x) and another angle (360-x) if i have to go the right say 1/4 to get to point how much do i need to go to the right to get to another point?

[icy jacinth]

uhmmm 1/4?

[ancient estuary]

why?

[icy jacinth]

Wait, I'm giving my all to elaborate

Am I right???

[ancient estuary]

you are... but i encourage you whether you an answer right or wrong always ask why ... understanding how you got the answer is more important

[icy jacinth]

Okay, if I'm right, then it's cuz we are moving the same amount from one point to another, right??.

[ancient estuary]

absolutely... but the reason should come first... so because we would move the same amount to the right to get to point and another point if we move 1/4 to right to get to point we will also move 1/4 to right to get to another point

[ancient estuary]

make sense?

[icy jacinth]

100%

[ancient estuary]

if it helps you can fold the circle into half to better see that you need to go the same amount 1 sec

[icy jacinth]

Alright

[ancient estuary]

o take the two halves and put them ont top of each other

[ancient estuary]

then you can see they are the same just on opposite sides on the circle

so we see the other patter that if we have an angle x and another angle 360-x then we will go to the right the same amount to get to their points
that means that cos(x) will give us the same answer as cos(360-x)

and we see our patter cos(x) = cos(360-x)

so try cos 50 and cos 310

cos 60 and coss 300

cos0 and cos 360

and so on and they should be the same does that make sense?

[icy jacinth]

Yup

[ancient estuary]

are you absolutely sure?

[icy jacinth]

Yes

[ancient estuary]

right so back to the question ha

we have cosx = 1/2

[icy jacinth]

Yeah

[ancient estuary]

from your calculator you got x = 60 right?

[ancient estuary]

from your calculator you got x = 60 right?

[icy jacinth]

Yup

[ancient estuary]

but because we are smart we know that cos(x) = cos(360-x) meaning cos60 = cos300

meaning we go to the right the same amount to get to our point when x is 60 and x is 300 make sense?

[icy jacinth]

Yeah

[ancient estuary]

this is why pi/3 is a solution and 5pi/3 (300) is also a solution

[ancient estuary]

and that is how we can easily also get that 5pi/3 is a solution

[icy jacinth]

Ohh

Okay, that's my "OPEN EYES"

[ancient estuary]

if we look at the circle we see that there are only two values of x that will make us go half to the right so when x is 60 and when x is 300

there are values that will make us go half to the left (like 120 and 240) but none other from 0 to 360 will make us go half to the right

does this make sense @icy jacinth

?

[icy jacinth]

Yes

[ancient estuary]

absolutely sure?

[icy jacinth]

Thonk

No

Hahaha

[ancient estuary]

no worries

remember our value of x from 0 to 360 will get us to a point on the circle

[ancient estuary]

then cos(x) tells us how much to the right we need to go to get to our point

from our calculator we know that when x is 60 we need to go 1/2 to the right ( because our calculator says cos60 is 1/2)

when x changes to 61 the amount we need to go to the right changes

[icy jacinth]

Okay, that makes sense

I think I follow

[ancient estuary]

if you look at the diagram you see that when x changes the amount you need to go to the right changes too

[icy jacinth]

Okay, yeah

[ancient estuary]

so when we get to 60 the amount we need to go the right is 1/2 but once we go passed 60 it changes and keeps on changing until the next time we need to go 1/2 to the right is when x is 300

[ancient estuary]

does this make sense @icy jacinth ?

[icy jacinth]

oH

I GOT IT

[ancient estuary]

absolutely sure?

[icy jacinth]

Okay I'll try to explain on my own words...

Sorry, it takes me a long time to think about how to explain it....
Uhm
So...
Let's say we got an x1 value...
so it has its...
Alter ego 😭😭
let's say x2
And let's say...
1/3 works for them...
Specifically at those points it works...
If we are on the journey to get from one point to another, the values change, right???

[ancient estuary]

that is great!! you should spend time thinking about explaining things yourself

[ancient estuary]

yes. i understand your reasoning though your phrasing may not be the most correct

[icy jacinth]

😭😭😭

Okay

[ancient estuary]

there is one pattern we didnt generalize though you knew it already

[ancient estuary]

we saw that adding 360 to an angle will get us to the same point on the circle

so cos(x) = cos(360+x)

does this make sense @icy jacinth ?

[icy jacinth]

Yeah, actually, I was testing the roots that we had already calculated...
And by doing that...
Example 7π/3
That would get me 420°
Which is 360° + 60°

[ancient estuary]

yes !!

so when x is 60 we go half to right , if we add 360 we get 420 so when x is 420 we also go 1/2 to the right and if we add 360 again ( 780) we also go 1/2 to the right

the same for when x is 300, 660, 1020

does that make sense @icy jacinth ?

so now we have a lot of values of x where cosx = 1/2

i will arrange them and use pi to define them

[icy jacinth]

ah yesss

[ancient estuary]

so pi/3, 5pi/3, 7pi/3, 11pi/3, 13pi/3, 17pi/3, 19pi/3, 23pi/3...

[icy jacinth]

Mhm

[ancient estuary]

do you have access to the full question?

[ancient estuary]

do you have access to the full question? as i think we might need to know more to get a full answer

for now i will assume n >= 1 or x > 60 as without those conditions it is near impossible to get an answer

[icy jacinth]

I don’t know how specific I can be to make myself clear, but… he just told us that he wants a formula that determines all the roots

he doesn't want this

[ancient estuary]

ill give you an incomplete answer for now then

[ancient estuary]

but alright we have a set of x values

when you have a set for an answer try to make the values you have look like the set formula for example we can factor out pi/3 from our values to make them look like the set answer we are supposed to have

so we have
pi/3(1, 5, 7, 11, 13, 17, 19, 23 .....)

make sense @icy jacinth ?

[icy jacinth]

yup

[ancient estuary]

is n supposed to start from 0 or 1?

[icy jacinth]

0

[ancient estuary]

hmmm....

that means for our for our "0th" term A + (-1)^0 should be 1
does that make sense @icy jacinth ?

[icy jacinth]

yup

[ancient estuary]

absolutely sure?

[icy jacinth]

yes

[ancient estuary]

so what should A be?

[icy jacinth]

0

[ancient estuary]

hmmm...

[icy jacinth]

😭

[ancient estuary]

so A + (-1)^0 = 1 ... what is anything to the power of 0 equal?

[icy jacinth]

1

[ancient estuary]

you are correct ha i just thought that -1 ^ 0 is -1 😭

teachers make mistakes too as long as what you do is logical dont be afraid to bring to notice the error to whoever it is

right so naively A should be 0 but that cant be right which is why im asking to see the full question

because when n is 1 our "1st term " will be pi/3 * (0 + (-1)^ 1) which is not equal to pi/3 * 5

however if we say n should start at 1 and we ignore pi/3 from our solutions so our values are 5pi/3, 7pi/3, 11pi/3, 13pi/3, 17pi/3, 19pi/3, 23pi/3

[ancient estuary]

we then have pi/3 * ( 5 , 7 , 11, 13, 17, 19, 23, ....)

so it will mean that A + (-1)^1 = 5
because for the first term of the set in the answer it will be pi/3 * ( A + (-1)^ 1)
and the first term in our values is pi/3 * (5) so A + (-1)^1 = 5

this will mean A = 6

@icy jacinth ignore the stuff that i put a line through

[icy jacinth]

okay

[ancient estuary]

and now we can see that A=6 works as for the second term we have pi/3 * (6 + (-1)^2)
= pi/3 * 7 which is our second term @icy jacinth

in short unless there are extra conditions im not sure we can find a value of A that works though i will think a bit more about it later

[icy jacinth]

oh I see

[ancient estuary]

does the above make sense?

[icy jacinth]

yeah

[ancient estuary]

are you absolutely sure?

[icy jacinth]

yup

[ancient estuary]

absolutely absolutely sure?

[icy jacinth]

100%

[ancient estuary]

great then !

i will think a little bit more about the problem to see if there is another answer for A but i dont think there is . I think the answer you are supposed to get it 6

phew... that was long

[icy jacinth]

ikr?? 😭

[ancient estuary]

It was my fault that it took so long but it was necessary

[icy jacinth]

oh np, tysm

I learned new stuff

[ancient estuary]

that is great to hear. this is how i encourage you and everyone to learn maths. break stuff down to the foundational things and make sure you understand why behind everythings

when you understand everything math then becomes fun... you arent just memorising stuff

even if i forget the patterns we talked about in an exam all i need to do is just draw the unit circle and because i understand how cos relates to the circle i can just get those patterns again quickly

there are many more patterns by the way ill encourage you to spend a bit exploring trig

but as you can see understanding everything can take a while and can be frustrating but when you understand it that is it

[icy jacinth]

yeah tbh that it was very interesting, I felt like I was in kindergarten and I laughed a lot
New Childhood Unlocked: Rediscovering Trigonometric Functions
You are very good at teaching 11/10

[ancient estuary]

thank you 🙂 i know i would have loved someone like me to teach me when i was younger... maths is very interesting when you understand all of it

and everyone can understand it... it might just take time

we shoudl also be patient with those that might be struggling

[icy jacinth]

Omg you are so nice 🐢🎀
You really encourage me to continue learning about this

[ancient estuary]

thank you ha... as i said this is the beginning of trig you will need to learn other patterns so you can be quick when solving trig equations...
note that when exploring these patterns it is best for your x value to be acute ( less than 90)
sin(180-x) = sinx
sin(180+x) = sinx
sin(360-x) = -sinx

cos(180-x) = -cosx

cos(180+x) = -cosx

cos(360-x) = cosx)

you should play around with the unit circle and see if those are true ( i should have put a few wrong ones in there to make sure you check ha i will put one wrong one there)

oof but then you will be able to see which one i edited

oh not really but you know it is among the top 4

from those one you can then get the patterns for tan too as we know tanx = sinx/cosx

[icy jacinth]

😭😭😭

It seems that you are very passionate about this

[icy jacinth]

sin (180+x) = -sinx
Isn't??

[ancient estuary]

is that correct or wrong and could you show me a diagram proving your answer?

[icy jacinth]

👁️ - 👁️

Let me take out my circle drawing equipment

[ancient estuary]

no need for circles to be perfect

im fortunate to have a drawing pad around me that is why my diagrams were neat my work is generally not super neat ha... and it doesnt usually need to be as long as your idea it gotten across

[icy jacinth]

Yeah I already gave up on the drawing app, so gimme a sec 😭😭😭

I am an expert at drawing circles, and anyone who says otherwise knows nothing about artistic appreciation

Now let me do the cos

[ancient estuary]

oh my goodness !!

insanely good work !!

yes ! sin(180+x) = sin(360-x) = -sinx

so if i see that sin30 = 1/2 i can immediately say that sin210 is -1/2 and sin330 is -1/2

this way we are able to solve trig equations way faster

[icy jacinth]

I think all the ones for cos are right

[ancient estuary]

great yes!! all the cos ones are correct

my only recommendation is more detail in your drawings

[icy jacinth]

😭😭
Aight, thanks

[ancient estuary]

something similar for cos too so you are absolutely sure what angles you are dealing with

but yes great. you learn quick...

you then get to the patterns for tanx

[ancient estuary]

you might want to go eat or sleep or do something else ha

i will just write a few things you can go over later

[icy jacinth]

Hahaha yeah, thank you, You are my savior
I'm gonna sleep, anyways, thank you again, you are so nice 11/10 ;))
Tysm

Gudnaito
Or idk
Whatever time it is, wherever you are

[ancient estuary]

its morning where i am ... let me know the solution your professor gives for the question
you are very welcome. i sadly cant save anyone though i know a guy ( shameless Jesus plug)

if there is anything to note is maths isnt difficult
just make sure you understand everything from the ground up and you will be able to build a big house

as for the tan pattans pun intended ...

[icy jacinth]

Aight, cya so have a nice day :))
Thanks for everything

[ancient estuary]

ill do one tan pattan ( repeated cos im not sure you got it the first time )
lets take tanx first
remember tanx = sinx / cosx
if you look at your circle you can see both sinx and cosx are positive ( you go up to get to your point when your angle is x so sinx is positive and you go to the right to get to your point so cosx is also positive) therefore tanx is positive when x is acute(between 0 and 90) (sinx and cosx are also positive for all values of x between 0 and 90 too)

lets say we go an amount (root3)/2 up to get to our point and we go 1/2 to the right to get to our point when x is some value

[ancient estuary]

when our angle is 180+x if you look at the circle youll see that we go down the same amount to get to the point at 180+x that we went up to get to our point at x
and we go left the same amount we went right

so if we went up (root3)/2 to get to our point at x we will go down (root3)/2 to get to our point at 180+x and if we went right 1/2 to get to our point at x we will go 1/2 to the left to get to our point at 180+x

this means that sin(180+x) will be negative (because we are going down) and cos(180+x) will be negative ( we are going left) so we have a negative number at the top and a negative number at the bottom for sin(180+x) / cos(180+x) so tan(180+x) = tanx

so from the diagram tanx will be positive b / positive a

but tan(180+x) will be negative a / negative b which is still the same as tanx

so one pattern is tanx = tan(180+x)

does this make sense?

if so try and find how tanx relates to tan(180-x) and tan(360-x) too

and show proof 🙂

quite confusing though let me know if you are confused anywhere

[icy jacinth]

I don't know if I get it right

And the answer...
It turns out it was just a system of equations

[ancient estuary]

you got it right !

but the tan ones are quite confusing make sure you absolutely understand them

then there is a shortcut

ACTS or CAST

To easily remember the patterns

if your angle is between 0 and 90 All of cos sin and tan are positivie

if your angle is in the second quadrant ( between 90 and 180) only Sin is positive ( that is why there is S there)

so it means
sin(180-x) = sinx
cos(180-x) = -cosx
tan(180-x) = -tanx

remember sin tells how much up or down you need to get to your point on the circle and the amount up you need to go for x is the same for angle 180-x so the pattern makes sense and you can use similar logic to justify the rest as we have previously done

so if i know sin30 is 1/2 i can easily say sin150 is also 1/2

if i know cos60 is half i can easily say cos120 is -1/2

if i know tan60 is root3 i can easily say tan120 is -root3

if your angle is in the third quadrant (between 180 and 270) only Tan is positive ( why the T)
so
sin(180-x) = -sinx
cos(180-x) = -cosx
tan(180-x) = tanx

ill encourage you to write out the relationships for the last quadrant using the fact onyl Cos is positive there

The other thing id encourage you to do is then solve quite a lot of basic trig questions. The more you solve the more second nature it will become.

[ancient estuary]

i blame the question ha

there was no indication that another term with n should be in the bracket