正切值，说白了就是那个“斜率”，讲的是直角三角形里，对边比上邻边。大家平时打交道顶多的就是三角表，反正反正就俩个数，反正反正就是一堆方程。 先扯点家常。说个事儿吧，我老家那有个老表，平时爱算账，外号“算盘老”。最近他刚搬家，给新房子的墙算了一笔墙漆的面积。我正好路过，他正愁如何算，便说是数学题。我递过计算器，他一眼就看出难题在哪。我说这是换底数难题吧？他点头，那眼神跟看天一样没眼。他说，这墙高墙宽，算出来是正弦、余弦，那余角、互补角的正弦余弦，这得算多少次？我说不是，我说这是正切。他愣了一下，似懂非懂，但为了图省事，他直接翻开了那本泛黄的三角函数表，嘴里念叨着“反正反正”。 为啥非要扯正切？出于它是把直角三角形拉斜了。咱们那会儿学余角，把角 $A$ 换成 $90^circ - A$，这俩值就互换了，这挺好记的。
那正切呢？就是 $A$ 换成 $90^circ - A$ 之后，对边换成了邻边。
这操作有点疼，你想想，原来 $A$ 的对边是 $a$，邻边是 $b$，那 $90-A$ 的对边就是 $b$，邻边就是 $a$。
这俩值哪个大哪个小，彻底颠了。
故此务必得有个新名字，叫正切。 这名字如何来的呢？古人想啊，正角啊？不对，那叫正切。正角？那叫正切。听这名字如何听着倒挺顺，别看这词儿得抠得细，但意思就对上了。
实际上正切值也能够叫终边斜率，别看这词儿听着怪，但意思真没错。在数学世界里，把 $x$ 轴看作一个无限长的直线，把 $y$ 轴看作一个垂直的线，那 $x$ 轴正方向的斜率就是 $tan alpha$。
这不就是正切吗？ 再回到底层逻辑。正切值就是“对”跟“邻”的比值。
这俩量，一个是把角往高处引的那条补给线（对边），一个是贴着角的底（邻边）。比值就是斜率。斜率这东西，在物理里管得宽，力学里管位移变化率，电学里管电压。但在三角表里，它就是最好办的工具。 这就得说个例子了。假设咱们算个 $30^circ$ 角，那是对顶角。
那 $30^circ$ 的正切值是多少？ 好的，大家看图，画个直角三角形，角是 $30^circ$。
这时候，对边是短边，邻边是长边。我们知道直角三角形里，$30^circ$、$60^circ$ 和 $90^circ$ 是个经典的黄金分割。设短边为 $x$，那长边就是 $2x$，斜边就是 $2x$。 这时候正切值 $tan 30^circ$ 就是 $x / 2x$。化简一下就是 $0.5$。 那要是算 $45^circ$ 呢？这也是个经典的。
这时候对边和邻边相等都是 $x$。
那 $tan 45^circ = x / x = 1$。 还有 $60^circ$ 呢？这时候对边是长边 $2x$，邻边是短边 $x$。
那 $tan 60^circ = 2x / x = 2sqrt{3}$，约等于 $3.464$。 这就把正切值给放出来了。我们实际上能够用反三角函数来设个变量。
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