Trigonometric functions and circular measure - the Australian ...

{16} • Trigonometric functions and circular measureSuppose ABC is a triangle and that the angles A and C are acute. Drop a perpendicularfrom B to the line interval AC and mark the lengths as shown in the following diagram.BchaAb – xDbxCIn the triangle ABD, applying Pythagoras’ theorem givesc 2 = h 2 + (b − x) 2 .Also, in the triangle BC D, another application of Pythagoras’ theorem givesh 2 = a 2 − x 2 .Substituting this expression for h 2 into the first equation and expanding,c 2 = a 2 − x 2 + (b − x) 2= a 2 − x 2 + b 2 − 2bx + x 2= a 2 + b 2 − 2bx.Finally, from triangle BC D, we have x = a cosC and soc 2 = a 2 + b 2 − 2ab cosC .This last formula is known as the cosine rule.Notice that, if C = 90 ◦ , then since cosC = 0 we obtain Pythagoras’ theorem, and so wecan regard the cosine rule as Pythagoras’ theorem with a correction term.The cosine rule is also true when the angle C is obtuse. But note that, in this case, thefinal term in the formula will produce a positive number, because the cosine of an obtuseangle is negative. Some care must be taken in this instance.By relabelling the sides and angles of the triangle, we can also write the cosine rule asa 2 = b 2 + c 2 − 2bc cos A and b 2 = a 2 + c 2 − 2ac cosB.