
This right triangle will allow us to to do right triangle trigonometry using SOH-CAH-TOA definitions. Those definitions are briefly explained here, and they are explained in much more depth in the
Right Triangle Trigonometry section in the Trigonometry Realms of Zona Land.
Here are the right triangle definitions for the sine, cosine, and tangent of an acute angle in a right triangle:Usually we just summarize these three definitions with these three short sentences:
The sine equals opposite over hypotenuse.
The cosine equals adjacent over hypotenuse.
The tangent equals opposite over adjacent.
Now, let's get back to our right triangle formed by the original vector and its x-component and y-component. Here's the diagram, fully labeled:

Notice that the x-component forms the side adjacent to the 35 degree angle, and that the y-component forms the opposite side to the 35 degree angle.
Let's find the size of the x-component; that is, let's find the size of the adjacent side.
We know the hypotenuse, (316 Newtons), and we know the angle, (35 degrees). We want to find the length of the adjacent side, (x-component). What trigonometry function relates the hypotenuse, an acute angle and its adjacent side in a right triangle? The cosine function does. The math looks this way:
Now, since the original vector is named F, its x-component is named Fx. This would be read 'F sub x'. So, in the above math we should remove 'x-component' and replace that term with Fx, as in:
We can solve for Fx by doing a little algebra and looking up the cosine of thirty-five degrees:
So, the x-component of the original vector is equal in size to 259 Newtons.
Now, realize this: The method for finding the x-component described here will not tell you the sign, (+ or -) for its value. This method will only tell you the size of the component. Notice that the x-component is pointing to the right. This makes it a positive x-component. (It would be negative if it pointed to the left.) So, we would would finally conclude that the x-component has a size of positive 259 Newtons.
Okay, now let's find the size of the y-component; that is, let's find the size of the opposite side. Again, we know the hypotenuse, (316 Newtons), and we know the angle, (35 degrees). We want to find the length of the opposite side, (y-component). It is the sine function that relates the hypotenuse, an acute angle and its opposite side in a right triangle. The math looks this way:
Now, since the original vector is named F, its y-component is named Fy which would be read 'sub y'. So, in the above math we should remove 'y-component' and replace that term with Fy, as in:
We can solve for Fy much like we solved for Fx: