Question

differentiate y=sin^-1[(5sinx + 4cosx)/√41]. i was stuck for 10 minutes on just this question in my test. it was so annoying!

Answer 1

we can simplify this
[(5sinx + 4cosx)/√41].

Answer 2

know the formula for
sin (A+B) ?

Answer 3

dont remember it right now...

Answer 4

:D

Answer 5

how can you use sin(a+b) here?

Answer 6

k, the formula is sin(a+b)=sin a . cos b + cos a . sin b

Answer 7

like this
[(5sinx + 4cosx)/√41]

5/√41 sin x + 4/√41 cos x
let 5/√41 be cos t
then 4/√41 = sin t

so we have
5/√41 sin x + 4/√41 cos x
= cos t sin x + sin t cos x
= (sin (x+t))
got this ??

Answer 8

how is 4/√41 = sin t ? is that part of the assumption ?

Answer 9

where t is some constant

Answer 10

assumption is " 5/√41 be cos t"

from this can u find sin t ?

Answer 11

uh, i dont see how you can get sin t from that.. do you use the identity sin^2+cos^2 ?
(i hope im not sounding dumb)

Answer 12

thats correct. use that identity
you'll actually get sin t = 4/√41

that was the point of taking cos t as 5/√41 (and not just cos t = 5 or something)

Answer 13

sin^2 t+cos^2 t =1

Answer 14

okkkK! thanx a ton!!! but will i always be able to find sin t as a constant already present in the question? in these kind of sums?

Answer 15

depends on the question actually, most of the times yes.

Answer 16

and in the end when i have to resubstitute t, will it be like this cos t = 5/sqrt41, hence, t = cos^-1(5/sqrt41) ?

Answer 17

since there is sqrt 41 term, i would say you can avoid it and simplify it further by finding tan t
tan t = 4/5 right ?
so you can put t= tan inverse (4/5)

anyways, in this case you won't even need to re-substitute

Answer 18

t = cos^-1(5/sqrt41) is also correct though

Answer 19

At the end of the simplification part, it looks something like this
y = sin^1. sin (x+t)
so the sin^1 and sin get cancelled,
and the eqn is
y= x + t
and when i differentiate it, i dont need to resubstitute t?

Answer 20

its a constant

Answer 21

so dy/dx will be just 1?

Answer 22

correct
't' is some constant, not dependng on x
so no need to re-substitute
yes. dy/dx = 1 is correct

Answer 23

ok! wow i dont believe the final answer could be so simple.. thanks a lot!

Answer 24

@hartnn You're very sure right?

Answer 25

but the sin^1 and sin do cancel each other out right?

Answer 26

yes they do cancel.

why @primeralph ? did you notice any error ?

Answer 27

@hartnn Not really; I didn't go over it at all. I just wanted to know if I should.

Answer 28

don't bother, its correct.

Answer 29

@Stew.a.r.t.
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Answer 30

k thnx