Trigonometric functions and circular measure - the Australian ...

{24} • Trigonometric functions and circular measureTo prove the cos(A − B) formula, from which we can obtain the other expansions, wereturn to the circle definition of the trigonometric functions.Consider two points P(cos A,sin A) and Q(cosB,sinB) on the unit circle, making anglesA and B respectively with the positive x-axis.yP (cos A,sin A)Q (cos B,sin B)A–1OB1xWe will calculate the distance PQ in two ways and then equate the results. First we applythe cosine rule to the triangle OPQ. Note that, in the diagram above, ∠POQ = A − B. Ingeneral, it is always the case that cos(∠POQ) = cos(A − B). So the cosine rule givesPQ 2 = 1 2 + 1 2 − 2 × 1 × 1 × cos(A − B) = 2 − 2cos(A − B).On the other hand, using the square of the distance formula from coordinate geometry,PQ 2 = (cosB − cos A) 2 + (sinB − sin A) 2= cos 2 A + sin 2 A + cos 2 B + sin 2 B − 2cos A cosB − 2sin A sinB= 2 − 2(cos A cosB + sin A sinB).Equating the two expressions for PQ 2 , we havecos(A − B) = cos A cosB + sin A sinB.We can easily obtain the formula for cos(A + B) by replacing B with −B in the formulafor cos(A −B) and recalling that cos(−θ) = cosθ (the cosine function is an even function)and sin(−θ) = −sinθ (the sine function is an odd function).Hencecos(A + B) = cos A cos(−B) + sin A sin(−B)= cos A cosB − sin A sinB.