URGENT SUMMER SCHOOL CALCULUS
posted by Derek on .
A force of magnitude 15 N is the resultant of two forces, one of which has a magnitude of 8 N and acts at an angle of 30° to the resultant. Find the magnitude and direction of the other force...how do I begin to solve this!?

Law of cosines:
The side you are looking for is opposite the 30 degree angle.
c^2=15^2+8^22*15*8cos30
I get a magnitude of about 9
For the angle, Iwould use law of sines. Draw a sketch first. 
Well, I am taking the resultant to be along the x axis and the 8 N vector at 30 degrees above the x axis.
Then our vector of length x will lie below the x axis an angle y
Now there are two things that have to be true
8 cos 30 + x cos y = 15
8 sin 30  x sin y = 0
or
6.93 + x cos y = 15
4  x sin y = 0
or
x cos y = 8.07
x sin y = 4
but we know sin y / cos y = tan y so divide
tan y 4/8.07
y = 26.4 degrees below x axis
which is
26.4 + 30 = 56.4 degrees from original vector
x = 4/sin26.4 = 9 Newtons 
You can also solve this problem purely algebraically without using geometry or drawing diagrams. If we denote the resultant force by R, the force of magnitude 8 N at an angle of 30° to the resultant by y and the unknown force by x, we have:
x + y = R >
x = R  y
Square both sides:
x^2 = R^2 + y^2  2 R dot y
The inner product R dot y can be expressed as:
R dot y = Ry cos(30°)
This then gives x = about 9 N
To find the angle of x to the resultant (let's call this theta), you can take the inner product of x with R and use that
x dot R = x R cos(theta)
We have:
x = R  y
taking the inner product of both sides with R gives:
x dot R = R^2  y dot R =
R^2 yRcos(30°)
If we divide both sides by xR we get cos(theta) on the l.h.s.:
cos(theta) = [R  ycos(30°)]/x
this gives theta = 26.4° 
My diagram doesn't work out, I have done it to scale, but I can't figure out how to draw it. I got the same answers as everyone above, I used the cosine and sine law to do mine (I got my answer after I posted :) ). But my diagram looks weird. Any thoughts?

15 cos

two forces of magnitude 10N and 30N act upon a body in direction making angle of 30 and 45 with x axis find the ressultant force?