#Integraition with trig identities

204 messages · Page 1 of 1 (latest)

vast hornet
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help me

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im insure why this guy used cos2A=cos^2A - sin^2A

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instead of cos^2A + sin^2A = 1

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then just rearrange for sin^2A = 1 - cos^2A

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<@&791435371564892232>

tepid otter
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that just very disgusting to do

vast hornet
tepid otter
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cos2a = 1- 2sin^2 a
sin^2 x = 1/2 (1 - cos2a)
this allows us to rewrite it with smth that has a cos with power of 1

vast hornet
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so we use that identity because it has no powers?

tepid otter
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ye

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and we can integrate cos(...) right

vast hornet
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yes it woiuld be sin

tepid otter
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ye

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so it makes it much easier to integrate

vast hornet
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cos^2A + sin^2A = 1

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what is the point of this

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like when do i use this

tepid otter
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normally when solving trig equations

vast hornet
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?

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what

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can you please give me an example when i would use each

tepid otter
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the cos2A is used mainly in trig integrals like
integrating cos^2 x

while the sin^2x + cos^2x one is used to solve trig equations
solve 5sin^2 x + 6cosx + 3 = 0

vast hornet
tepid otter
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let me rephrase everything i said

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we dont know how to integrate sin^2 (...) or cos^2(...) right

vast hornet
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yea

tepid otter
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so we wanna try rewrite these in a way such that its either sin(...) or cos(...)

vast hornet
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yeah

tepid otter
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there is no way to do that with sin^2(x) + cos^2(x) = 1

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right?

vast hornet
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yes that is true

tepid otter
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but with cos(2a) = 1-2 sin^2(a) and cos(2a) = 2 cos^2(a) - 1
we can rearrange for sin^2(a) or cos^2(a)

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and it will have smth that is in terms of cos(2a)

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which we can integrate

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u get that?

vast hornet
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wait

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uhhh

tepid otter
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cos(...) we know how to integrate right?

vast hornet
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what is this

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where is the 1 coming ffrom

tepid otter
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they rewrote the top one using the cos^2 + sin^2 = 1

vast hornet
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what the

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so this guy used 2 identities

tepid otter
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ye

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the cos double angle and the sin^2 and cos^2 one

vast hornet
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this is so confusing

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how did he get 2cos^"A

tepid otter
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cos^2 (x) - sin^2 (x)
cos^2(x) - (1 - cos^2 (x))
2 cos^2(x) - 1

vast hornet
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what will cos2A equal to

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ok nvm

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uhh

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okok so cos2A = 2cos^2A -1

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yeah

tepid otter
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we would rearrange that for cos^2 x

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and the substitute that in with the relavent replacement of x

vast hornet
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wait

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how am i getting here

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where is the sin coming from

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we only have cos so far

tepid otter
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cos^2 (x) - sin^2 (x)
(1 - sin^2 (x)) - sin^2 (x)
1 - 2 sin^2(x)

vast hornet
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im at cos2A=2cos^2A - 1

tepid otter
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2 cos^2(x) - 1
2(1 - sin^2 (x)) - 1
1 - 2sin^2(x)

vast hornet
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ok yeah

tepid otter
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we rearrange that for sin^2 (x)

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sin^2(a) = 1/2 (1-cos2a)

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we can substitute that in

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so the integral becomes
1/2 - 1/2 cos2x

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and we can integrate that

vast hornet
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why has bro done 1/2sin2x

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why 1/2

tepid otter
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when we integrate cos(2x)
we can do this thru a u-sub
u = 2x
du = 2 dx
dx = 1/2 du
so it becomes
1/2 cos(u)

integrating that gives 1/2 sin(u) = 1/2 sin(2x)

vast hornet
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what

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is du the inside of the backet and dx the outside

tepid otter
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let me write this down and do it

vast hornet
tepid otter
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have u done integration by substitution?

vast hornet
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nope

tepid otter
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ah

vast hornet
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tf is this man i hate integration

tepid otter
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what website it that?

vast hornet
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i hate anything with trig

vast hornet
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okok how did you decide to use integration by substution and not part of something idk

tepid otter
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u done this?

image.png
vast hornet
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yeh

tepid otter
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that would have given u the reason why

vast hornet
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oh damn wait

tepid otter
tepid otter
vast hornet
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ookok

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i really need to remmeber this

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ok

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tf just happened

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id like to say he did 2x everuthing but he didnt

tepid otter
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basically he did

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but since we dont wanna change the value

vast hornet
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2x1/2 =1

tepid otter
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we also multiply by 1/2

vast hornet
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what

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why

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why would we ever want to do x1/2

tepid otter
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we dont want to

vast hornet
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bro we just did

tepid otter
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but kinda had to caus we multiplied the bracket by 2

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and we dont wanna change the value

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tbf, that first line would be fine as an answer

vast hornet
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naw i wanna know what he did

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this guy is crazy

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how he get 1/4 if we do x2

tepid otter
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he times everything in the bracket by 2

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and multiplied the fraction by 1/2

vast hornet
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you say dont wanna change the value but idkw hat that means

vast hornet
tepid otter
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exactly
we multiplied by 1 in the end

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so the value does not change

vast hornet
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dude if i did this in the exam my final answer would be 2x-sin2x + c

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not 1/4(

tepid otter
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i would just expand the bracket and call it a day with this

vast hornet
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im losing my mind

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how many marks would i get for this question?

tepid otter
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i think like 6-7

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let me check rq

vast hornet
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thats cold asf

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q1: integrate sin^2x

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so small yet heavy

tepid otter
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lol

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is this as or a2?

vast hornet
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huh

tepid otter
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like is this first yr or second yr content?

vast hornet
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this is alevel pure integration with trig identities

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idk y12 i did all of pure

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its probably 2nd yr

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Could you please look at this

IMG_8781.jpg
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Back to

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When do I use cos^2A + sin^2A = 1

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And when do I use Cos2A=cos^2A - sin^2A

tepid otter
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normally, other than in trig identity questions

vast hornet
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Like what kinda question because for this one we in start with that

tepid otter
vast hornet
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We used the rearranged double angle formula

tepid otter
vast hornet
vast hornet
tepid otter
vast hornet
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That has a power

tepid otter
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u could use sin^2 + cos^2 = 1 to get it all into an equation in terms of cos

vast hornet
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So do we use the 2nd identity

tepid otter
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no

vast hornet
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Huhh

tepid otter
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when trying to solve trig equation, we wanna keep everything in terms of cos(x) or sin(x)

vast hornet
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Yeah I get that

tepid otter
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we dont wanna introduce a 2x inside the cos

vast hornet
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Why

tepid otter
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then it just becomes ugly

vast hornet
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I’m losing my mind

tepid otter
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probs best to get some sleep lol

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and come back to this tmrw

vast hornet
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So we use this to?????

IMG_8782.png
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We use this when there is only 1 squared thing

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We use this if there is

IMG_8783.png
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Sh what

tepid otter
tepid otter
vast hornet
vast hornet
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Ohhh

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Yeah go on

tepid otter
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tbf, getting used to which one to use when is kinda like how u would have decided which circle theorem to use

vast hornet
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Circle theorem?

tepid otter
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in gcse

vast hornet
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Oh

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How does that relate

tepid otter
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like, in gcse, u would often try to spot a certain pattern which makes it feel like u must use a certain theorem

vast hornet
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I miss gcse math

vast hornet
tepid otter
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it comes with practice

vast hornet
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I am by far the worst at series and sequences, differentiation, integration, anything with trig

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I got a long way to go

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Stats, mech, everything else has been kinda smooth

tepid otter
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but ye:
use cos^2 + sin^2 = 1
when we want it to be in terms of cos(x) or sin(x) and we dont care what the power is
like in trig equations, since we can classify it as a hidden polynomial which we just solve

use cos2a
when we want to get rid off the power completely
like in integration, since we dont know how to integrate sin^2 or cos^2

vast hornet
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Thank you

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I am a slow learner

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But this exam solutions guy solved this problem in less than a minute

tepid otter
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lol

vast hornet
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I’ll ask my teacher too just to calcify

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Clarify*

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Thanks

tepid otter
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np

vast hornet
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@tepid otter

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Can I ask

vast hornet
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Or is it just specifically these two

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?

tepid otter
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they apply to those only really