Do we need to memorise this identity? (1 Viewer)

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016
Solving the following integral:



I subbed x=tan(theta)
t=atan(theta/2)
 
Last edited:

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
Solving the following integral:

Which identity? You will need to leave the final answer in terms of x.

For that integral, you could also use a trig. substitution of x = a*tan(theta).

You end up with something involving integral of cosec(theta), which is:

-ln(cosec(theta) + cot(theta)).

Then you could convert this back to in terms of x, using trig. identities or a right-angled triangle to assist you. This avoids use of tan(theta/2).
 
Last edited:

leehuan

Well-Known Member
Joined
May 31, 2014
Messages
5,805
Gender
Male
HSC
2015
Rule of thumb with sqrt(a2+x2) is to sub x=a.tan(theta)

No memorising here aside from that^
 

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016
So my qs is

put tan(theta/2) in terms of x where x=tan(theta)
 

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016


This identity isn't expected to be memorised for the HSC, but you should be able to derive it if needed.
Could derive it, but imagine me sitting in the 4U trials thinking

x=tan(theta)
t=tan(theta/2)

"Oh boy better derive that weird squareroot cosine formula!!"

haha so is there any clue in the question or solution that allows me to conclude that formula is needed

Assuming I do derive this in the exam, the integral is still quite tedious

1.png
2.png
3.png
 
Last edited:

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
Could derive it, but imagine me sitting in the 4U trials thinking

x=tan(theta)
t=tan(theta/2)

"Oh boy better derive that weird squareroot cosine formula!!"

haha so is there any clue in the question or solution that allows me to conclude that formula is needed
Well we didn't need that formula to do that integral in the original post. We could avoid half-angles as I showed. But if you did introduce half-angles, the way to get out of them is to use the half-angle formulas in some way, because these formulas tell us how to go from sin(a/2) and cos(a/2) (and hence tan(a/2)) to trig functions of a. (That's how I derived that formula above.)
 

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016
Which identity? You will need to leave the fnal answer in terms of x.

For that integral, you could also use a trig. substitution of x = a*tan(theta).

You end up with something involving integral of cosec(theta), which is:

-ln(cosec(theta) + cot(theta)).

Then you could convert this back to in terms of x, using trig. identities or a right-angled triangle to assist you. This avoids use of tan(theta/2).
Seem to be getting
Any way I can avoid half angle results?

See attached images in my previous post
(sorry I understand my explanation of my issue is not too great)
 

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016
Ohh ok - Will make sure to remember that :)
Teacher told me its not a standard integral, so its best to use t-formula
If you dont mind, can you show me how I can finish off the solution from
Given t=tan(theta/2)

(just curious thats all)
 

leehuan

Well-Known Member
Joined
May 31, 2014
Messages
5,805
Gender
Male
HSC
2015
T-formulae does allow the integral of cosec to come out nice and tidy, but there is no need for it at all
 

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
Ohh ok - Will make sure to remember that :)
Teacher told me its not a standard integral, so its best to use t-formula
If you dont mind, can you show me how I can finish off the solution from
Given t=tan(theta/2)

(just curious thats all)






 

SammyT123

Active Member
Joined
Nov 16, 2014
Messages
360
Gender
Male
HSC
2016
Thankyou once again Integrand
You legend

Was wondering how I can stop leeching off the BOS community and give something back LOL
Probably not in the maths section, will look forward to helping out the physics and chemistry students tho :D
(after hscs :awesome:)
 

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,556
Location
Outside reality
Gender
Male
HSC
2016


This will convert the integral into a standard form (look in your booklet bro)

The answer very rapidly comes out to be

 

InteGrand

Well-Known Member
Joined
Dec 11, 2014
Messages
6,109
Gender
Male
HSC
N/A
Ohh ok - Will make sure to remember that :)
Teacher told me its not a standard integral, so its best to use t-formula
If you dont mind, can you show me how I can finish off the solution from
Given t=tan(theta/2)

(just curious thats all)
 

Paradoxica

-insert title here-
Joined
Jun 19, 2014
Messages
2,556
Location
Outside reality
Gender
Male
HSC
2016
T-formulae does allow the integral of cosec to come out nice and tidy, but there is no need for it at all
Refer to my previous post.

Also half the time a trig substitution is unnecessary for indefinite integrals.

But for definite integrals they are very slick.
 

leehuan

Well-Known Member
Joined
May 31, 2014
Messages
5,805
Gender
Male
HSC
2015
Refer to my previous post.

Also half the time a trig substitution is unnecessary for indefinite integrals.

But for definite integrals they are very slick.
It's only tidy, if you have cosec by itself. (And it's still not necessary either, but it makes the final answer tidy)
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top