The Laws of Sines and Cosines are not fancy ideas. No! They are about the triangular geometies of things.
Crank, Rod and Piston
These mechanical parts of single-cylinder gasoline engines work together to transform combustion energy to locomotion. We identify the centerline of the crank as a stationary origin, 0. The letters A and B identifuy the locations of the piston and wrist pin, respectively.
Engine designers must know the position of the piston, 0A, for every position of the crank, 0B. A vector diagram for the components is shown. The operating variables are the angles, α and β. The vector lengths are the variable length, a(t), and the constant lengths, b, and c. Describe the geometry of the engine components with vectors. Reduce the vectors to obtain two algebraic equations that relate the angles and lengths of the mechanism.
♦ We start with this vector equation:
0A = 0B + BA
It is obvious that the above statement is true (for the position shown and for all other positions. Next we write the above implicit equation, explicitly. We write vectors for which the length is known (and constant) in magnitude-direction form.
aK = [ c sinα I + c cosα K]
+ [ b sinβ (- I) + b cosβ K]
It is important that this equation be correct. Double check it! Next combine the right-side I and K components.
a K = [ c sinα - b sinβ] I + [ c cosα + b cosβ ] K
Although we know which variables in the above equation vary in time, let us rewrite the equation to show its time dependence - explicitely.
a(t) K = [ c sinα(t) - b sinβ(t)] I + [ c cosα(t) + b cosβ(t) ] K
This shows the time-dependent variables to be a(t), α(t) and β(t). Time and much writing can be avoided by not writing the dependence, "( t )". Now we vector scalar multiply this equation first by the unit vector I, then by K.
The I - component multiplication yields: 0 = c sinα - b sinβ. This rearranges to be the Law of Sines:
sinα/b = sinβ/c
The K component multiplication yields: a = c cosα + b cosβ or (a - b cosβ) = c cosα
Next, we take the I and K relations and square them:
I : (b sinβ)2 = (c sinα)2 and K: ( a - b cosβ)2 = ( c - cosα)2
Now add these squares:
a2 - 2 a b (cosβ) + b2 (cosβ)2 = c2(cosα)2
b2 (sinβ)2 = c2 (sinα)2
With a touch of algebra the result is the Law of Cosines:
a2 - 2ab(cosβ) + b2 = c2
Students who have studied trigonometry (but not vectors) would have considered using these two laws. Some might remark, why bother with vectors? Vectors are powerful. Trigonometry is built into them. Also, if time dependence is introduced, as say α(t), the vector equations differentiate readily to yield the relationships of component velocities.