Trigonometric & Polar Functions
AP Precalculus Topic 3 8:08 English narration · English + 中文 subtitles burned in
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Transcript
Watch a point travel around a circle at a steady rate.
看一个点以稳定的速度绕圆运动。
Its height rises, falls, and rises again, over and over.
它的高度上升、下降，再上升，一次又一次。
Trace that height against time, and you get a smooth, repeating wave — a sine wave.
把这个高度随时间画出来，你就得到一条平滑、重复的波——正弦波。
This is periodic motion: a Ferris wheel, a sound, the tides, the seasons.
这就是周期运动： 摩天轮、声音、潮汐、四季。
Trigonometry is the mathematics of things that repeat.
三角学，就是研究"重复"的数学。
Welcome to Unit Three: trigonometric and polar functions.
欢迎来到第三单元：三角与极坐标函数。
We build the sine and cosine from the unit circle, shape them into sinusoids, meet tangent and the inverse functions, and finish with a new way to plot — polar coordinates.
我们从单位圆建立正弦和余弦，把它们塑造成正弦型函数， 认识正切和反函数，最后学习一种新的作图方式——极坐标。
Let's begin.
让我们开始吧。
Everything starts on the unit circle — a circle of radius one, centered at the origin.
一切都从单位圆开始——一个半径为一、以原点为中心的圆。
Take any angle in standard position — vertex at the origin, initial ray on the positive x-axis — and follow the terminal ray to where it meets the circle.
取任意一个处于标准位置的角—— 顶点在原点，始边落在正 x 轴上——沿着终边找到它与圆的交点。
Measure the angle in radians: the arc length cut off on that unit circle.
用弧度来量这个角： 弧度就是它在单位圆上截出的弧长。
The exact values at the special angles come from two triangles you already know, the isosceles right triangle and the equilateral triangle, with the signs set by the quadrant.
特殊角处的精确值来自两个你已经熟悉的三角形—— 等腰直角三角形和等边三角形——正负号则由角所在的象限决定。
The cosine of the angle is simply the x-coordinate of that point.
这个角的余弦，就是该点的 x 坐标。
The sine of the angle is the y-coordinate.
这个角的正弦，就是 y 坐标。
And the tangent is the ratio, y over x — the slope of the ray.
而正切是这个比值，y 比 x——也就是射线的斜率。
Three functions, all read straight off one circle.
三个函数，都直接从一个圆上读出。
Now let the angle grow, and plot the results.
现在让角增大，把结果画出来。
Sine starts at zero and rises — it is the height on the circle.
正弦从零开始上升——它是圆上的高度。
Cosine starts at one and falls — it is the horizontal position.
余弦从一开始下降—— 它是水平位置。
Both oscillate smoothly between minus one and one, repeating every full turn.
两者都在负一和一之间平滑起伏，每转一整圈就重复一次。
Look closely: they are the same wave, one just shifted from the other by a quarter turn.
仔细看： 它们是同一条波，只是彼此相差四分之一圈。
Stretch and shift a sine wave, and you get a sinusoidal function.
把正弦波拉伸、平移，就得到一般的正弦型函数。
Four numbers control it.
四个数控制它。
The amplitude, the size of a, is how far it reaches above and below its center.
振幅，也就是 a 的大小， 是它相对中心上下能到多远。
The period, two pi over b, is how long one full cycle takes; its reciprocal is the frequency, how many cycles fit in one unit.
周期，二派除以 b，是一个完整循环需要多久；它的倒数就是频率， 也就是一个单位里能装下多少个循环。
The midline, d, is the center line the wave swings around.
中线 d，是波所围绕的中心线。
And c shifts it sideways: a plus c shifts the wave to the left.
而 c 把它左右平移：括号里加上 c，波就向左移。
So for three sine of two theta, plus one: amplitude three, period pi, midline one.
所以对于"三乘正弦二theta加一"：振幅三，周期派，中线一。
Fit a sinusoid to real data — tides, daylight, temperature — and it is only trustworthy over its contextual domain, the stretch of inputs the data actually covers.
用正弦型函数去拟合真实数据——潮汐、日照、气温——它只在自己的上下文定义域内可信， 也就是数据真正覆盖的那段输入范围。
The tangent function is sine over cosine — the slope of that rotating ray.
正切函数是正弦除以余弦——就是那条旋转射线的斜率。
As the ray turns, its slope repeats every half turn, so tangent has period pi.
射线旋转时，它的斜率每半圈重复一次， 所以正切的周期是派。
But where cosine is zero, the slope becomes infinite: the graph shoots up to a vertical asymptote and reappears from the bottom.
但在余弦为零的地方，斜率变成无穷大：图像冲向一条竖直渐近线， 再从底部重新出现。
Between asymptotes, tangent always increases.
在相邻渐近线之间，正切始终递增。
Can we run trig backwards — give a ratio, get the angle?
我们能把三角函数倒着用吗——给一个比值，求出角？
Not quite, because the functions repeat, so many angles share one value.
不完全行，因为这些函数会重复， 许多角共享同一个值。
The fix is to restrict the domain — keep just one piece of the sine, from minus pi over two to pi over two.
办法是限制定义域——只保留正弦的一段，从负二分之派到二分之派。
On that piece it is one-to-one, so it has an inverse: the arcsine.
在这一段上它是一一对应的，于是有反函数：反正弦。
Cosine and tangent get the same treatment, giving arccosine and arctangent.
余弦和正切也这样处理， 就得到反余弦和反正切。
It takes a ratio and returns an angle, and its graph is the restricted sine reflected across the line y equals x.
它接受一个比值，返回一个角， 它的图像是那段正弦关于直线 y 等于 x 的反射。
Now solve a trig equation.
现在解一个三角方程。
Two sine theta equals one, over one full turn.
二乘正弦 theta 等于一，在一整圈范围内。
Divide by two: sine theta equals one half.
除以二：正弦 theta 等于二分之一。
On the unit circle, draw the line at height one half — it crosses the circle in two places, in the first and second quadrants.
在单位圆上，画出高度为二分之一的水平线——它与圆相交于两处，在第一和第二象限。
So there are two solutions: theta is pi over six, or five pi over six.
所以有两个解：theta 等于六分之派，或六分之五派。
And because sine repeats, over all real numbers there are infinitely many — just add multiples of two pi.
又因为正弦会重复，在全体实数上有无穷多个—— 只要加上二派的整数倍。
Three more functions round out the set — the reciprocals.
还有三个函数使这一套完整——倒数函数。
Secant is one over cosine.
正割是余弦分之一。
Cosecant is one over sine.
余割是正弦分之一。
And cotangent is one over tangent.
余切是正切分之一。
Each blows up to an asymptote wherever its bottom hits zero — secant where cosine is zero, cosecant where sine is zero.
每一个都会在分母为零的地方冲向渐近线——正割在余弦为零处， 余割在正弦为零处。
The unit circle also hands us identities.
单位圆还给了我们恒等式。
Drop the two legs from a point on the circle: the horizontal leg is cosine, the vertical leg is sine, and the radius is one.
从圆上一点放下两条直角边：水平边是余弦，竖直边是正弦，半径是一。
By the Pythagorean theorem, those two legs squared add to one.
由勾股定理，这两条边的平方相加等于一。
That gives the most important identity in trigonometry: sine squared plus cosine squared equals one.
这就给出三角学中最重要的恒等式： 正弦平方加余弦平方等于一。
From it flow all the more identities — the sum, difference, and double-angle formulas.
由它推出所有其余的恒等式——和角、差角与二倍角公式。
Those identities are worth writing out, because the exam quotes them.
这些恒等式值得完整写出来，因为考试会直接引用它们。
The sine of alpha plus beta is sine alpha, cosine beta, plus cosine alpha, sine beta.
alpha 加 beta 的正弦，等于 sin alpha 乘 cos beta，加上 cos alpha 乘 sin beta。
The cosine version looks similar but hides a trap: cosine alpha, cosine beta, MINUS sine alpha, sine beta.
余弦的版本看起来类似，但藏着一个 陷阱：cos alpha 乘 cos beta，减去 sin alpha 乘 sin beta。
Sine keeps the plus; cosine flips to a minus.
正弦保持加号；余弦变成减号。
For the difference formulas, flip every sign that sits in front of a beta term.
对于差角公式，把每一个 beta 项前面的符号都反过来。
Now the useful trick: set beta equal to alpha, and the sum formulas collapse into the double-angle identities.
现在是有用的技巧：令 beta 等于 alpha，和角公式就坍缩成二倍角恒等式。
Sine of two alpha is two sine alpha cosine alpha.
二 alpha 的正弦等于 二 sin alpha cos alpha。
Cosine of two alpha is cosine squared alpha, minus sine squared alpha.
二 alpha 的余弦等于 cos 平方 alpha 减 sin 平方 alpha。
And because sine squared plus cosine squared is one, that last one can be written three different ways — in cosines only, or in sines only.
而因为 sin 平方加 cos 平方等于一， 最后这一个可以写成三种不同的形式——只用余弦，或者只用正弦。
Which form you choose is usually what makes a problem easy or hard.
选择哪一种形式，往往 决定了一道题是容易还是困难。
Here is a different way to name a point.
这是给一个点命名的另一种方式。
Instead of across and up, use distance and direction.
不用"横向和纵向"，而用"距离和方向"。
Polar coordinates give the distance r from the origin, and the angle theta of the ray to the point.
极坐标给出 到原点的距离 r，以及指向该点的射线的角 theta。
To convert to the usual coordinates, x is r cosine theta, and y is r sine theta.
要化成普通坐标，x 等于 r 乘余弦 theta， y 等于 r 乘正弦 theta。
And backward, r is the square root of x squared plus y squared.
反过来，r 等于 x 平方加 y 平方的平方根。
A complex number, x plus y i, is a point in exactly the same way, so it can be written with r and theta too.
复数 x 加 y i 也正是同样的一个点，所以它同样可以用 r 和 theta 写出来。
Once a point is set by r and theta, we can graph an equation that links them: r equals a function of theta.
一旦一个点由 r 和 theta 确定，我们就能画出把它们联系起来的方程：r 等于 theta 的某个函数。
As theta sweeps around, the radius grows and shrinks, and the point traces a curve.
当 theta 扫过一圈，半径时大时小，点就描出一条曲线。
This one, r equals one plus cosine theta, sweeps out a heart-shaped curve called a cardioid.
这一条，r 等于一加余弦 theta， 扫出一条心形曲线，叫做心脏线。
Circles, spirals, roses, and limaçons all come from simple polar rules.
圆、螺线、玫瑰线和蚶线，都来自简单的极坐标规则。
Watch r as theta increases: while r is growing the curve moves away from the origin, and while r is shrinking it moves toward it.
注意 theta 增大时 r 怎么变：r 在变大时，曲线远离原点；r 在变小时，曲线靠近原点。
Where it switches, the distance hits a relative extreme.
在它由增变减、或由减变增的地方，这个距离取到相对极值。
The average rate of change of r with respect to theta — delta r over delta theta — says how fast it moves in or out.
r 关于 theta 的平均变化率——delta r 除以 delta theta——就说明它向外或向内移动得有多快。
Before you go, three things to remember.
结束之前，三个要点。
First, live on the unit circle: cosine and sine are the coordinates, always between minus one and one.
第一，扎根于单位圆：余弦和正弦就是坐标，永远在负一和一之间。
Second, for a sinusoid, read off the amplitude, the period two pi over b, the midline, and the phase shift.
第二，对于正弦型函数，读出振幅、周期二派除以 b、中线，以及相移。
Third, convert between polar and rectangular with x equals r cosine theta, and y equals r sine theta.
第三，用 x 等于 r 乘余弦 theta、y 等于 r 乘正弦 theta，在极坐标和直角坐标之间转换。
Master these, and Unit Three is yours.
掌握这些，第三单元就是你的了。