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I'm having difficulty understanding when to use $\cos$ and $\sin$ to find $x$ and $y$ components of a vector. Do we always use $\cos$ for $x$-component or what?

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    $\begingroup$ Draw the problem. Find the triangles. See where the angle is (this may involve applying a little Euclidean geometry). $\endgroup$ Mar 13, 2015 at 20:42
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    $\begingroup$ $\cos$ is always associated with the adjacent side. $\sin$ is always associated with the opposite side. That's all I ever remember, and luckily that's all that's needed. $\endgroup$
    – BMS
    Mar 13, 2015 at 20:56

2 Answers 2

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It depends on your definition of the angle:

enter image description here

In the picture as drawn, $x$ is $r\cos\alpha$ and $y$ is $r\sin\alpha$. But if I chose a different convention for $x$, $y$ or $\alpha$ I would need a different equation.

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One thing I've found useful is having a mental picture of the $\sin$ and $\cos$ functions. If you draw your vector and split it up into components like so:

enter image description here

and pick your angle $\theta$, then you can vary that angle in your mind and see what happens to the components.

If you let $\theta$ go to zero, all of the vector will be in the x-direction, so taking $F_x = |F| \alpha \hat e_x$ you'll see that we need to set $\alpha = 1$ ($\alpha$ is just a factor that varies between -1 and 1).

So at $\theta = 0$ we have $\alpha = 1$. Now redo this with $\theta = \pi/2$ and you'll see that there is no component in the x-direction, our $\alpha$ will be 0.

We now need to find a function of $\theta$ that returns 1 for $\theta = 0$ and 0 for $\theta = \pi/2$. Looking at this graph

enter image description here

we see that the $\cos(\theta)$ does just that. The x-component of $\vec F$ is therefore given by $F_x = |F| \cos(\theta) \hat e_x$.

By doing the same again but this time watching out what happens to the y-component, we will see that the $\sin$ gives you the right relation between angle and scaling factor.

So, without memorizing SOH CAH TOA or something like that, you can figure out how the components are given.

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