My Vector Enigma: Solving Analytically w/ Trig Functions

[Noesis]

My question follows off of the picture attached.

I have been able to answer both questions, but I still have questions of my own.

I had to solve them geometrically using the Law of Cosines and the Law of Sines...why is it that I am not able to break it into components using the trig functions?

Surely it has something to do with the axes not being perpendicular..but what exactly?

As far as the u axis goes...it's just a direction, independent of v, so I don't understand why I cannot solve it using trig functions multiplied by the magnitude.

The answer for the second problem, is about 205 N via the Law of Sines:

Fx = (300*sin(40))/sin(110)

Why can't I simply do cos(30)*300 ?

And when I try to implement a normal perpendicular system, I still get the wrong components. What am I doing wrong?

And how would you solve this analytically with trig functions and not geometrically?

Thank you so much guys.

 

[Noesis]

Also if I shift the bottom axis over 20 degrees to make it perpendicular, and try cos(25)*500 to get the y component, and cos(30)*300 to get the x, then use both to get the magnitude, I actually get a different magnitude than I would have using the Law of Cosines.

It's a bit less.

Not understanding why either.

 

[kyle8921]

I can offer some insight into your questions. It is important to understand that there are different ways to solve a problem, and some methods may be more appropriate or efficient depending on the situation. In this case, using the Law of Cosines and the Law of Sines is a valid method of solving the problem geometrically. However, it may not be the most efficient or accurate way to solve it analytically using trigonometric functions.

To answer your first question about breaking the problem into components using trig functions, it is important to note that trigonometric functions only work for right triangles where one of the angles is 90 degrees. In your problem, the axes are not perpendicular, so the triangle formed is not a right triangle. This means that the trigonometric functions cannot be used directly to solve for the components.

For your second question, using cos(30)*300 assumes that the angle between the force and the u axis is 30 degrees, which may not be the case. The Law of Sines takes into account the angles and sides of a non-right triangle, making it a more accurate method of solving the problem.

To solve the problem analytically with trig functions, you can use the law of cosines to find the magnitude of the force and then use the law of sines to find the angles and components. Alternatively, you can use vector addition and trigonometric functions to find the components of the force in the u and v directions.

In summary, it is important to understand the limitations and assumptions of different methods when solving a problem. In this case, using the Law of Cosines and the Law of Sines is a valid method of solving the problem geometrically, but may not be the most efficient or accurate way to solve it analytically using trigonometric functions.