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From: Fred Nurk on 6 Jul 2010 20:29
If z = sin(x) and sin(1) = a then, using the first order approximation,
the value of sin(1.1) is equal to:

A 0.1(1 - a ^ 2) ^ (1 / 2)
B 0.1cos(1)
C 0.1(1 - a ^ 2) ^ (1 / 2) + a
D a + 0.1
E 0.1a

Here's my working out and what I got up to:
sin(1 + 1 / 10) approximately = sin(1) + (1 / 10)f'(1)
approximately = a + (1 / 10)f'(1)
d(sin(x)) / dx = cos(x)
cos(1) =

From here, I can say that C is the only answer given that could be right.
But is it impossible to write cos(1) in terms of a and/or z? Am I able to
show that sin(1.1) is C?

TIA,
Fred
From: Francois Grieu on 7 Jul 2010 02:02
Le 07/07/2010 02:29, Fred Nurk a écrit :
> If z = sin(x) and sin(1) = a then, using the first order approximation,
> the value of sin(1.1) is equal to:
>
> A 0.1(1 - a ^ 2) ^ (1 / 2)
> B 0.1cos(1)
> C 0.1(1 - a ^ 2) ^ (1 / 2) + a
> D a + 0.1
> E 0.1a
>
> Here's my working out and what I got up to:
> sin(1 + 1 / 10) approximately = sin(1) + (1 / 10)f'(1)
> approximately = a + (1 / 10)f'(1)
> d(sin(x)) / dx = cos(x)
> cos(1) =
>
> From here, I can say that C is the only answer given that could be right.
> But is it impossible to write cos(1) in terms of a and/or z? Am I able to
> show that sin(1.1) is C?


Hint: sin(a)^2 + cos(a)^2 = 1. Check the sign of sin(a).

Francois Grieu
From: Francois Grieu on 7 Jul 2010 03:08
Le 07/07/2010 02:29, Fred Nurk a écrit :
> If z = sin(x) and sin(1) = a then, using the first order approximation,
> the value of sin(1.1) is equal to:
>
> A 0.1(1 - a ^ 2) ^ (1 / 2)
> B 0.1cos(1)
> C 0.1(1 - a ^ 2) ^ (1 / 2) + a
> D a + 0.1
> E 0.1a
>
> Here's my working out and what I got up to:
> sin(1 + 1 / 10) approximately = sin(1) + (1 / 10)f'(1)
> approximately = a + (1 / 10)f'(1)
> d(sin(x)) / dx = cos(x)
> cos(1) =
>
> From here, I can say that C is the only answer given that could be right.
> But is it impossible to write cos(1) in terms of a and/or z? Am I able to
> show that sin(1.1) is C?


Hint: sin(1)^2 + cos(1)^2 = 1. Check the sign of cos(1).

Francois Grieu
[reposted with correction]
From: Fred Nurk on 7 Jul 2010 05:00
Francois Grieu wrote:
> <snip>
> Hint: sin(1)^2 + cos(1)^2 = 1.
Right. cos(1) = {1 - [sin(1)] ^ 2} ^ (1 / 2)
= (1 - a ^ 2) ^ (1 / 2)

> Check the sign of cos(1).
Why? It is now clear that the answer is C.

> <snip>

TIA,
Fred
From: Francois Grieu on 7 Jul 2010 06:21
Fred Nurk wrote:
> If z = sin(x) and sin(1) = a then, using the first
> order approximation, the value of sin(1.1) is equal to:
>
> A 0.1(1 - a ^ 2) ^ (1 / 2)
> B 0.1cos(1)
> C 0.1(1 - a ^ 2) ^ (1 / 2) + a
> D a + 0.1
> E 0.1a

> Francois Grieu wrote:
>> Hint: sin(1)^2 + cos(1)^2 = 1.
> Right. cos(1) = {1 - [sin(1)] ^ 2} ^ (1 / 2)
> = (1 - a ^ 2) ^ (1 / 2)
>
>> Check the sign of cos(1).
> Why? It is now clear that the answer is C.

Because on the problem

If z = sin(x) and sin(2) = a then, using the first
order approximation, the value of sin(2.1) is equal to:
A 0.1(1 - a ^ 2) ^ (1 / 2)
B 0.1cos(2)
C 0.1(1 - a ^ 2) ^ (1 / 2) + a
D a + 0.1
E 0.1a

the right answer would be: neither of the above,
due to a sign issue.


Note: in the schooling system I practiced, there was no
multiple choices tests, one had to give the reasoning,
and would not get the highest mark without stating all
relevant hypothesis used.


Francois Grieu
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