# When to use $\sin$ and $\cos$ to find $x$,$y$ components?

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?

• Draw the problem. Find the triangles. See where the angle is (this may involve applying a little Euclidean geometry). Mar 13, 2015 at 20:42
• $\cos$ 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.
– BMS
Mar 13, 2015 at 20:56

It depends on your definition of the angle:

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:

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

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.