How Do Magnetic Fields Interact at Point P Between Parallel Wires?

[GDGirl]

Homework Statement

Two long thin parallel wires a distance d = 15.3 cm apart carry 25-A currents (I) in the same direction. The magnetic field is measured at point P, a distance d1 = 13.3 cm from the lower wire and a distance d2 = 5.3 cm from the upper wire.

a. Find the x and y components of the magnetic field at P due to the current in the lower wire only.
HELP: First find the magnitude of the field due to the current in the lower wire.
HELP: To find the direction of the field, the Law of Cosines will be helpful: for a triangle of sides a, b, and c, the angle θ between sides a and b is given by c² = a² + b² - 2 a b cosθ.

https://wug-s.physics.uiuc.edu/cgi/courses/shell/common/showme.pl?cc/Knox/phys130a/spring/homework/13/04/HW14_4.jpg

Homework Equations

c² = a² + b² - 2 a b cosθ.
Magnitude=\mu₀I/2\pir

The Attempt at a Solution

So I found the magnitude to be \mu₀(25)/2\pi(.133)=3.759E-5
then I use the Law of Cosines to find that theta=19.814
so then 90-19.814=70.186
Cos of that= y/3.759e-5
which gives me y=1.274e-5
Which is correct.
However, when I use Sin of 70.186=x/3.759e-5 I get 3.536e-5, but this is not correct. I'm not sure what I'm doing wrong here. :/ I've never quite understand how to tell when you use Sin and Cos for which value anyway.

 

[Redbelly98]

Your work looks pretty good, so I imagine the problem is one of the following:

Wrong number of significant figures in final answer,
Final answer slightly wrong from roundoff error,
or
+/- Sign error in final answer.

In your diagram, have you drawn the direction of B due to the lower wire? That might help clear up whether it's + or -.

 

[GDGirl]

I didn't even think about the fact that it could be negative. B is going anti-clockwise, so it's negative. Thanks!