inverse trig problem (1 Viewer)

[jimmik]

this is jus a simple question im sure but ive jus started self teaching myself inverse trig ahead of the class and i need a lil help. the ques is as follows:

without using tables or a calculator, prove that:
integrate 1/(1+x^2)dx between x=3/5 and x=4 is equal to pi/4

 

[SamD]

This is a standard integral so...

LHS=arctan x from 3/5 to 4
= arctan 4 - arctan 3/5

Let u=arctan 4
so tan u =4

Let v=arctan 3/5
so tan v=3/5

Sort of looks like we can use the expansion for tan(a-b) using u and v rather than a and b

Which is....

tan(u-v)=(tan u + tan v)/(1- tanu tanv)

Substituting tanu=4 and tanv=3/5 gives...
tan(u-v)=(4+3/5)/(1-4.3/5)
=1

Therefore, u-v=pi/4 (As tan pi/4=1)

Therefore, arctan 4 - arctan 3/5 =pi/4

HTH
Sam

 

[fatmuscle]

whoa, not only is he SDD King, but Maths King too

 

[SamD]

I was a Maths teacher also... however it's been more than 10 years since I ever looked at any 3Unit Maths stuff. My daughter's doing the HSC this year, so I'm also doing a bit of a brush up!

Sam

 

[Ragerunner]

Does your daughter do SDDand IPT?

 

[KeypadSDM]

How do you prove Tan[Pi/4] = 1

Without using a calculator or standard tables?

(I'm assuming a standard table is what is derived from the usual triangles)

 

[Affinity]

standard tables are computed using series expansions.

proving tan(pi/4)=1 depends on what definitions are used

 

[Grey Council]

wouldn't you just draw up that triangle thing?

draw a rightangled isosceles triangle, with sides 1, 1 and root 2. Tan45 = 1

I think i'm missing something here...

EDIT:
oh, can't use the "standard tables" aka triangles. right. thought i was doing something wrong. hehe

 

[EliteMasterPro]

prove tan pi/4 = 1 by using GENERAL TRIG SOLUTIONS

 

[SamD]

No she doesn't do SDD or IPT, apart from MSN messenger, computers really don't interest her in the slightest! She is using this site, her username is Louisey!, so help her out, particularly with English. I haven't got a clue about how to help her with English!

I think it's pretty reasonable to assume tan(pi/4)=1 is understood, I wouldn't bother trying to prove it. Similarly with the tan(a-b) expansion and the standard integral.

HTH
Sam

 

[KeypadSDM]

Originally posted by EliteMasterPro
prove tan pi/4 = 1 by using GENERAL TRIG SOLUTIONS

Which isn't allowed by the question.

But the question in itself is a bit rich. You can't assume any values for Tan that are derived from the triangle.

Maybe working out Tan(1 - 1/3 + 1/5 - 1/7 + ...) might give you something ...

 

[Affinity]

define tan(x), then I will give you a proof

 

[Grey Council]

lol @ samd and his daughter. hahahaha, good one.

 

[m3h_m3h]

easiest way is to draw the damn graphs

then u will see

 

[jimmik]

thanks for all the input. im learning it away slowly hehe. cheers!

 

[KeypadSDM]

Originally posted by m3h_m3h
easiest way is to draw the damn graphs

then u will see

Yeah, but the question asks you not to do it with them