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?

  • 5
    $\begingroup$ 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
  • 2
    $\begingroup$ $\cos$ 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


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.


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .