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The cosine rule gives the resultant magnitude, but it does not fully give the direction. In vector problems, an answer such as is incomplete if the force direction is not stated.
For two vectors, the resultant direction can be found using the sine rule. This rule works on the resultant triangle formed by the two vectors and their resultant.
Here, is the included angle between and , is the resultant magnitude, and is the angle between and resultant .
The sine rule is usually used after the resultant magnitude is known.
Before calculating, decide the reference direction. If the question asks for the resultant direction relative to , the required angle is from toward .
If the question asks for direction relative to , the angle is different. Do not swap angles without checking the vector triangle.
A common relationship is:
This formula finds the resultant angle relative to vector . Because the side opposite angle has magnitude , appears in the numerator.
Two forces act at one point. , , and the included angle is . From the cosine rule, the resultant magnitude is .
Now find angle between and .
Therefore:
So the resultant force is about at from toward .
The same direction can be checked using components. If is placed on the positive axis, then:
For the same example:
This gives , matching the sine rule. This check is useful when you are unsure which angle is being found.
The sine rule is less practical when there are more than two vectors or when each vector direction is already given relative to coordinate axes. In those cases, the component method is usually safer because all directions are organized along the and axes.
Use the sine rule when the problem is clear: two vectors, a known included angle, a known resultant magnitude, and a requested resultant direction relative to one of the vectors.
