Vector problem , algebraic and trigonometric complications

Kamhogo

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Joined
Jan 4, 2016
Messages
28
Hi. This originally a physics problem, but it's the mathematical
part that's giving me a headache.

The problem:
Vector E(net) = 2Ecos(theta)
E= Kq/[(x^2) + 0.25(s^2)] where K,q and s are constants
Theta= Tan^(-1)(0.5s/x)
Calculate Vector E(net)

My work:

1) cos (tan^(-1)x)=1/(1+x^2)^(1/2)
Cos(tan^(-1)(0.5s/x))= 1/[1+(0.25s^(2)/x^(2))]^(1/2)
= 1/[(x^(2)+0.25s^(2))/x^(2)]^(1/2)
= 1/[x*(x^(2)+0.25s^(2))^(1/2)]

2)E(net)= [2Kq/{(x^(2)+0.25s^(2)}]*[1/{x*(x^(2)+0.25s^(2))^(1/2)}]
= 2Kq/[x*(x^(2)+0.25s^(2))^(3/2)]



But my textbook says the answer is E(net)= 2Kqx/(x^(2)+0.25s^(2))^(3/2)

What am I doing wrong?
 
Last edited:
Hi. This originally a physics problem, but it's the mathematical
part that's giving me a headache.

The problem:
Vector E(net) = 2Ecos(theta)
E= Kq/[(x^2) + 0.25(s^2)] where K,q and s are constants
Theta= Tan^(-1)(0.5s/x)
Calculate Vector E(net)

My work:

1) cos (tan^(-1)x)=1/(1+x^2)^(1/2)
Cos(tan^(-1)(0.5s/x))= 1/[1+(0.25s^(2)/x^(2))]^(1/2)
= 1/[(x^(2)+0.25s^(2))/x^(2)]^(1/2)

= x/[(x^(2)+0.25s^(2))^(1/2)]

= 1/[x*(x^(2)+0.25s^(2))^(1/2)]

2)E(net)= [2Kq/{(x^(2)+0.25s^(2)}]*[1/{x*(x^(2)+0.25s^(2))^(1/2)}]
= 2Kq/[x*(x^(2)+0.25s^(2))^(3/2)]



But my textbook says the answer is E(net)= 2Kqx/(x^(2)+0.25s^(2))^(3/2)

What am I doing wrong?

Right there....
 
Hi. This originally a physics problem, but it's the mathematical
part that's giving me a headache.

The problem:
Vector E(net) = 2Ecos(theta)
E= Kq/[(x^2) + 0.25(s^2)] where K,q and s are constants
Theta= Tan^(-1)(0.5s/x)
Calculate Vector E(net)

My work:

1) cos (tan^(-1)x)=1/(1+x^2)^(1/2)
Cos(tan^(-1)(0.5s/x))= 1/[1+(0.25s^(2)/x^(2))]^(1/2)
= 1/[(x^(2)+0.25s^(2))/x^(2)]^(1/2)
= 1/[x*(x^(2)+0.25s^(2))^(1/2)]

2)E(net)= [2Kq/{(x^(2)+0.25s^(2)}]*[1/{x*(x^(2)+0.25s^(2))^(1/2)}]
= 2Kq/[x*(x^(2)+0.25s^(2))^(3/2)]



But my textbook says the answer is E(net)= 2Kqx/(x^(2)+0.25s^(2))^(3/2)

What am I doing wrong?
When you have mistakes like this (or even just want to 'check your answer' by working through the problem again), it is sometimes helpful to substitute for recurring expressions. For example, if we let
A = Kq
B = (x^2) + 0.25(s^2)
then we have
E(net) = 2Ecos(theta)
E= A/B where A is constant and B a function of x
Theta= Tan^(-1)(0.5s/x)

So
Theta= Tan^(-1)(0.5s/x) \(\displaystyle \Rightarrow\, cos(\theta)\, =\, \frac{x}{B^{\frac{1}{2}}}\)
So
E(net) = 2 (A/B)(x/B1/2) = 2 A x / B3/2
=2 Kq x / [(x^2) + 0.25(s^2)]3/2
 
Hi. This originally a physics problem, but it's the mathematical
part that's giving me a headache.

The problem:
Vector E(net) = 2Ecos(theta)
E= Kq/[(x^2) + 0.25(s^2)] where K,q and s are constants
Theta= Tan^(-1)(0.5s/x)
Calculate Vector E(net)

My work:

1) cos (tan^(-1)x)=1/(1+x^2)^(1/2)
Cos(tan^(-1)(0.5s/x))= 1/[1+(0.25s^(2)/x^(2))]^(1/2)
= 1/[(x^(2)+0.25s^(2))/x^(2)]^(1/2)
= 1/[x*(x^(2)+0.25s^(2))^(1/2)]

2)E(net)= [2Kq/{(x^(2)+0.25s^(2)}]*[1/{x*(x^(2)+0.25s^(2))^(1/2)}]
= 2Kq/[x*(x^(2)+0.25s^(2))^(3/2)]



But my textbook says the answer is E(net)= 2Kqx/(x^(2)+0.25s^(2))^(3/2)

What am I doing wrong?
Oh, and another thing. I have generally found it useful to think of inverse trig function in their 'original forms'. So when we have
\(\displaystyle \theta=tan^{-1}\frac{C}{D}\)
then 'the opposite' is C, 'the adjacent' is D, and thus the hypotenuse, H, is (C2+D2)1/2. Of course C and D can be scaled by any non zero S which would scale H by |S|. One must be somewhat careful in doing this: If the expression (C/D) is negative, assign C the negative value if you are looking for answers [\(\displaystyle \theta\)] in the first and second quadrant or assign D the negative value if you are looking for answers in the fourth and first quadrant.
 
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