In the analysis of statics problems, it’s often valuable to convert forces into equivalent force systems.  There are a number of reasons for this, as you’ll see in later chapters.  One of the most important conversions is that where a force is broken into components along the x- and y-axis.  This is called resolution of forces, and is illustrated graphically:
 

 

In the next tutorial, you will use this technique to find resultant forces quickly and easily.  As for computation, as long as an angle is specified between the force and an axis, resolution is accomplished simply with elementary trigonometry.

Consider the 200 lb. force shown.  The 36 degree angle is shown with respect to the x-axis.  A right triangle is drawn in which shows the x and y component of this force.
 

 

Remembering that the cosine of an angle is defined as the adjacent leg of a right triangle divided by the hypotenuse, it’s easy to see that
 

Likewise, the sine of an angle is the opposite leg divided by the hypotenuse.  Therefore,
 

It’s important to observe the direction of these forces and assign proper signs (positive or negative).  In the example above, because of the direction of the force, it’s easy to see that its projections on the x- and y-axis are both in the positive direction.  Therefore both F_x and F_y are positive.  Look at the drawings below.  You should see clearly why the component forces are either negative or positive in each example.
 

The vast majority of problems you’ll encounter in the text define angles for the forces with respect to the x-axis as you’ve seen in the examples thus far.  As a result, it’s common to get used to relating the x-direction component of a force with the cosine of the angle and the y-direction component with the sine of the angle.  However, if a force and angle is defined like this:

and you don’t use the proper trig relationships, you will not get the correct answers.  In this case:
 

 

The little “mini” triangle is intended to tell you what the missing angle is!  Imagine what this triangle would look like if it was enlarged:
 

It should be obvious now that the angle you need is found by using the tangent relationship (an angle’s tangent is defined by the opposite leg of the triangle divided by the adjacent leg):
 

When you’re doing resolution problems, a little common sense observation will usually show you if you’ve done the right thing.  Look at this force:
 

If you mentally sketch the x and y components you should see that the x component should be larger in magnitude than the y component.  Therefore, if your calculations tell you that F_x = 24 lb. and F_y = 97 lb., you should intuitively know that something went wrong!!  Look again at your math.  The vast majority of errors in statics problems, once you know the fundamentals of how they're solved, will be simple math mistakes!  These kind of common sense evaluations will help you catch and fix those mistakes.

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