第9章三角関数の恒等式と方程式Trigonometric Identities and Equations

• 9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions • 9.2 Sum and Difference Identities • 9.3 Double-Angle, Half-Angle, and Reduction Formulas • 9.4 Sum-to-Product and Product-to-Sum Formulas • 9.5 Solving Trigonometric Equations

・9.1 三角関数の恒等式の証明と式の整理・9.2 加法定理・9.3 倍角・半角・次数を下げる公式・9.4 和積・積和の公式・9.5 三角方程式を解く

When we think of tennis as a game of angles, we may imagine players racing up to the net, creating options to deliver powerful cross shots that will leave their opponent stumbling toward the line. This is an exciting and effective method of play, though it brings greater risk.

テニスを「角度の駆け引き」として見ると、ネットへ詰めた選手が鋭いクロスショットを放ち、相手をサイドラインへ走らせる場面が思い浮かぶ。見応えがあり、有効な攻め方だが、リスクも伴う。

But while the excitement of the game interplays with all types of geometry, some of the newest innovations make even more use of mathematics. With balls traveling well over 100 miles per hour judges cannot always discern the centimeter or millimeters of difference between a ball that is in or out of bounds. Professional tennis was among the first sports to rely on an advanced tracking system called Hawk-Eye to help make close calls. The system uses several high-resolution cameras that are able to monitor and the ball's movement and its position on the court. Using the images from several cameras at once, the system's computers use trigonometric calculations to triangulate the ball's exact position and, essentially, turn a series of two-dimensional images into a three-dimensional one. Also, since the ball travels faster than the cameras' frame rate, the system also must make predictions to show where a ball is at all times. These technologies generally provide a more accurate game that builds more confidence and fairness. Similar technologies are used for baseball, and automated strike-calling is under discussion.

テニスの戦術にはさまざまな幾何学的関係が関わっているが、新しい判定技術では数学がさらに直接的に使われる。時速100マイルを大きく超えるボールについて、ラインの内外を分ける数センチ、数ミリの差を、審判が常に見極めるのは難しい。プロテニスは、際どい判定を補助する追跡システム「ホークアイ」を早くから採用した競技の1つである。複数の高解像度カメラでボールの動きとコート上の位置を撮影し、コンピューターが三角法の計算によって位置を割り出す。複数の二次元画像から、三次元空間での位置を求めるわけだ。撮影と撮影の間にもボールは移動するため、その間の位置を推定する処理も必要になる。こうした技術は判定の精度を上げ、公平さと判定への信頼を支える。原書が書かれた時点では、野球でも類似技術が使われ、ストライク判定の自動化が議論されていた。