剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

三角函数与极坐标函数

AP 微积分预备 · 第 3 主题

训练
讲义 词汇表
3.1

周期现象
大纲
Learning ObjectiveEssential Knowledge

3.1.A
Construct graphs of periodic relationships based on verbal representations.

  • 3.1.A.1 A periodic relationship can be identified between two aspects of a context if, as the input values increase, the output values demonstrate a repeating pattern over successive equal-length intervals.
  • 3.1.A.2 The graph of a periodic relationship can be constructed from the graph of a single cycle of the relationship.

3.1.B
Describe key characteristics of a periodic function based on a verbal representation.

  • 3.1.B.1 The period of the function is the smallest positive value $k$ such that $f(x + k) = f(x)$ for all $x$ in the domain. Consequently, the behavior of a periodic function is completely determined by any interval of width $k$.
  • 3.1.B.2 The period can be estimated by investigating successive equal-length output values and finding where the pattern begins to repeat.
  • 3.1.B.3 Periodic functions take on characteristics of other functions, such as intervals of increase and decrease, different concavities, and various rates of change. However, with periodic functions, all characteristics found in one period of the function will be in every period of the function.

来源:美国大学理事会 AP 课程与考试说明

一个关系是周期性(periodic)的,若它的输出模式在规则的输入步重复。周期(period)是使所有 $x$ 满足 $f(x+k)=f(x)$ 的最小正 $k$。你能通过复制一个单一的周期段(cycle)构建整个图象,并通过找到模式重复的间隔多远来估计周期。在每个周期段内一个周期函数仍有递增/递减和极大值/极小值的区间。

sin 和 cos 在 -1 和 1 之间起伏;tan 在 90 和 270 度断裂
sin 和 cos 在 -1 和 1 之间起伏;tan 在 90 和 270 度断裂
词汇表 训练
英文 中文 拼音
periodic 周期性 zhōu qī xìng
period 周期 zhōu qī
cycle 周期段 zhōu qī duàn
练习卷
3.2

正弦、余弦与正切
大纲
Learning ObjectiveEssential Knowledge

3.2.A
Determine the sine, cosine, and tangent of an angle using the unit circle.

  • 3.2.A.1 In the coordinate plane, an angle is in standard position when the vertex coincides with the origin and one ray coincides with the positive $x$-axis. The other ray is called the terminal ray. Positive and negative angle measures indicate rotations from the positive $x$-axis in the counterclockwise and clockwise direction, respectively. Angles in standard position that share a terminal ray differ by an integer number of revolutions.
  • 3.2.A.2 The radian measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin subtended by the angle to the radius of that same circle. For a unit circle, which has radius $1$, the radian measure is the same as the length of the subtended arc.
  • 3.2.A.3 Given an angle in standard position and a circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The sine of the angle is the ratio of the vertical displacement of $P$ from the $x$-axis to the distance between the origin and point $P$. Therefore, for a unit circle, the sine of the angle is the $y$-coordinate of point $P$.
  • 3.2.A.4 Given an angle in standard position and a circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The cosine of the angle is the ratio of the horizontal displacement of $P$ from the $y$-axis to the distance between the origin and point $P$. Therefore, for a unit circle, the cosine of the angle is the $x$-coordinate of point $P$.
  • 3.2.A.5 Given an angle in standard position, the tangent of the angle is the slope, if it exists, of the terminal ray. Because the slope of the terminal ray is the ratio of the vertical displacement to the horizontal displacement over any interval, the tangent of the angle is the ratio of the $y$-coordinate to the $x$-coordinate of the point at which the terminal ray intersects the unit circle; alternately, it is the ratio of the angle's sine to its cosine.

来源:美国大学理事会 AP 课程与考试说明

一个角处于标准位置(standard position),当它的顶点在原点而它的始边位于正 $x$ 轴上时。它的弧度(radian)度量是一个单位圆上的弧长。对于终边遇到一个半径 $r$ 的圆的点 $P$:

$$\cos\theta = \frac{x}{r},\qquad \sin\theta = \frac{y}{r},\qquad \tan\theta = \frac{y}{x}\ (\text{the slope of the terminal ray}).$$

从角命名一个直角三角形的边
从角命名一个直角三角形的边
探索

Read sine and cosine off the unit circle

On the unit circle a point at angle $\theta$ has coordinates $(\cos\theta,\ \sin\theta)$. Spin the angle to watch sine and cosine trace out as the height and width.

词汇表 训练
英文 中文 拼音
standard position 标准位置 biāo zhǔn wèi zhì
radian 弧度 hú dù
练习卷
3.3

正弦与余弦函数值
大纲
Learning ObjectiveEssential Knowledge

3.3.A
Determine coordinates of points on a circle centered at the origin.

  • 3.3.A.1 Given an angle of measure $\theta$ in standard position and a circle with radius $r$ centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The coordinates of point $P$ are $(r\cos\theta, r\sin\theta)$.
  • 3.3.A.2 The geometry of isosceles right and equilateral triangles, while attending to the signs of the values based on the quadrant of the angle, can be used to find exact values for the cosine and sine of angles that are multiples of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{6}$ radians and whose terminal rays do not lie on an axis.

来源:美国大学理事会 AP 课程与考试说明

在一个半径 $r$ 的圆上,$x=r\cos\theta$ 而 $y=r\sin\theta$。特殊角处($0,\tfrac{\pi}{6},\tfrac{\pi}{4},\tfrac{\pi}{3},\tfrac{\pi}{2},\dots$)的精确值来自等腰直角和等边三角形(equilateral triangle)几何,符号由角的象限(quadrant)设定。

词汇表 训练
英文 中文 拼音
equilateral triangle 等边三角形 děng biān sān jiǎo xíng
quadrant 象限 xiàng xiàn
练习卷
3.4

正弦与余弦函数图像
大纲
Learning ObjectiveEssential Knowledge

3.4.A
Construct representations of the sine and cosine functions using the unit circle.

  • 3.4.A.1 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The sine function, $f(\theta) = \sin\theta$, gives the $y$-coordinate, or vertical displacement from the $x$-axis, of point $P$. The domain of the sine function is all real numbers.
  • 3.4.A.2 As the input values, or angle measures, of the sine function increase, the output values oscillate between $-1$ and $1$, taking every value in between and tracking the vertical distance of points on the unit circle from the $x$-axis.
  • 3.4.A.3 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The cosine function, $f(\theta) = \cos\theta$, gives the $x$-coordinate, or horizontal displacement from the $y$-axis, of point $P$. The domain of the cosine function is all real numbers.
  • 3.4.A.4 As the input values, or angle measures, of the cosine function increase, the output values oscillate between $-1$ and $1$, taking every value in between and tracking the horizontal distance of points on the unit circle from the $y$-axis.

来源:美国大学理事会 AP 课程与考试说明

单位圆画出正弦曲线

在单位圆(unit circle)($r=1$)上,$\sin\theta$ 是 $y$ 坐标而 $\cos\theta$ 是 $x$ 坐标。随着 $\theta$ 增加,两者以周期 $2\pi$ 在 $-1$ 和 $1$ 之间平滑地振荡(oscillate)。正弦从 $0$ 开始(上升);余弦从 $1$ 开始。它们是移动了 $\tfrac{\pi}{2}$ 的同一个波。

y = sin x 和 y = cos x 是 -1 和 1 之间的平滑波
y = sin x 和 y = cos x 是 -1 和 1 之间的平滑波
探索

Graph a sine wave

Unrolling the circle gives the wave $y=\sin x$: it repeats every $2\pi$ (its period) and swings between $-1$ and $1$ (its amplitude).

词汇表 训练
英文 中文 拼音
unit circle 单位圆 dān wèi yuán
oscillate 振荡 zhèn dàng
练习卷
3.5

正弦型函数
大纲
Learning ObjectiveEssential Knowledge

3.5.A
Identify key characteristics of the sine and cosine functions.

  • 3.5.A.1 A sinusoidal function is any function that involves additive and multiplicative transformations of $f(\theta) = \sin\theta$. The sine and cosine functions are both sinusoidal functions, with $\cos\theta = \sin\left(\theta + \dfrac{\pi}{2}\right)$.
  • 3.5.A.2 The period and frequency of a sinusoidal function are reciprocals. The period of $f(\theta) = \sin\theta$ and $g(\theta) = \cos\theta$ is $2\pi$, and the frequency is $\dfrac{1}{2\pi}$.
  • 3.5.A.3 The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. The amplitude of $f(\theta) = \sin\theta$ and $g(\theta) = \cos\theta$ is $1$.
  • 3.5.A.4 The midline of the graph of a sinusoidal function is determined by the average, or arithmetic mean, of the maximum and minimum values of the function. The midline of the graphs of $y = \sin\theta$ and $y = \cos\theta$ is $y = 0$.
  • 3.5.A.5 As input values increase, the graphs of sinusoidal functions oscillate between concave down and concave up.
  • 3.5.A.6 The graph of $y = \sin\theta$ has rotational symmetry about the origin and is therefore an odd function. The graph of $y = \cos\theta$ has reflective symmetry over the $y$-axis and is therefore an even function.

来源:美国大学理事会 AP 课程与考试说明

一个正弦型函数(sinusoidal function)是正弦(或余弦)的任何变换。它的关键特征:

  • 周期和频率(frequency)互为倒数;$\sin\theta$ 有周期 $2\pi$。
  • 振幅(amplitude)= 最大和最小输出之间距离的一半。
  • 中线(midline)$y=d$ = 最大和最小的平均(水平的中心线)。

图象每半个周期段交替上凹和下凹。

Worked example. 对于 $f(\theta)=3\sin(2\theta)+1$:振幅是 $3$、周期是 $\dfrac{2\pi}{2}=\pi$,而中线是 $y=1$。所以最大输出是 $1+3=4$ 而最小是 $1-3=-2$。

词汇表 训练
英文 中文 拼音
sinusoidal function 正弦型函数 zhèng xián xíng hán shù
frequency 频率 pín lǜ
Amplitude 振幅 zhèn fú
Midline 中线 zhōng xiàn
练习卷
3.6

正弦型函数的变换
大纲
Learning ObjectiveEssential Knowledge

3.6.A
Identify the amplitude, vertical shift, period, and phase shift of a sinusoidal function.

  • 3.6.A.1 Functions that can be written in the form $f(\theta) = a\sin(b(\theta + c)) + d$ or $g(\theta) = a\cos(b(\theta + c)) + d$, where $a, b, c,$ and $d$ are real numbers and $a \neq 0$, are sinusoidal functions and are transformations of the sine and cosine functions. Additive and multiplicative transformations are the same for both sine and cosine because the cosine function is a phase shift of the sine function by $-\dfrac{\pi}{2}$ units.
  • 3.6.A.2 The graph of the additive transformation $g(\theta) = \sin\theta + d$ of the sine function $f(\theta) = \sin\theta$ is a vertical translation of the graph of $f$, including its midline, by $d$ units. The same transformation of the cosine function yields the same result.
  • 3.6.A.3 The graph of the additive transformation $g(\theta) = \sin(\theta + c)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal translation, or phase shift, of the graph of $f$ by $-c$ units. The same transformation of the cosine function yields the same result.
  • 3.6.A.4 The graph of the multiplicative transformation $g(\theta) = a\sin\theta$ of the sine function $f(\theta) = \sin\theta$ is a vertical dilation of the graph of $f$ and differs in amplitude by a factor of $|a|$. The same transformation of the cosine function yields the same result.
  • 3.6.A.5 The graph of the multiplicative transformation $g(\theta) = \sin(b\theta)$ of the sine function $f(\theta) = \sin\theta$ is a horizontal dilation of the graph of $f$ and differs in period by a factor of $\left|\dfrac{1}{b}\right|$. The same transformation of the cosine function yields the same result.
  • 3.6.A.6 The graph of $y = f(\theta) = a\sin(b(\theta + c)) + d$ has an amplitude of $|a|$ units, a period of $\left|\dfrac{1}{b}\right|2\pi$ units, a midline vertical shift of $d$ units from $y = 0$, and a phase shift of $-c$ units. The same transformations of the cosine function yield the same results.

来源:美国大学理事会 AP 课程与考试说明

正弦函数的振幅、周期与相位

一般的正弦型函数是

$$f(\theta)=a\sin\big(b(\theta+c)\big)+d \quad\text{or}\quad a\cos\big(b(\theta+c)\big)+d,$$
其中 $|a|$ 是振幅、$\dfrac{2\pi}{|b|}$ 是周期、$-c$ 是水平相移(phase shift)(括号里的 $+c$ 把图像向左移 $c$),而 $d$ 是竖直移动(vertical shift)(中线)。一个负的 $a$ 反射这个波。

探索

Stretch and shift a sinusoid

In $y=a\sin(bx+c)+d$: $a$ sets the amplitude, $b$ the period, $c$ a horizontal phase shift, and $d$ a vertical shift. Change each and watch the wave respond.

词汇表 训练
英文 中文 拼音
phase shift 相移 xiāng yí
练习卷
3.7

正弦型函数的情境与数据建模
大纲
Learning ObjectiveEssential Knowledge

3.7.A
Construct sinusoidal function models of periodic phenomena.

  • 3.7.A.1 The smallest interval of input values over which the maximum or minimum output values start to repeat, that is, the input-value interval between consecutive maxima or consecutive minima, can be used to determine or estimate the period and frequency for a sinusoidal function model.
  • 3.7.A.2 The maximum and minimum output values can be used to determine or estimate the amplitude and vertical shift for a sinusoidal function model.
  • 3.7.A.3 An actual pair of input-output values can be compared to pairs of input-output values produced by a sinusoidal function model to determine or estimate a phase shift for the model.
  • 3.7.A.4 Sinusoidal function models can be constructed for a data set with technology by estimating key values or using sinusoidal regressions.
  • 3.7.A.5 Sinusoidal functions that model a data set are frequently only useful over their contextual domain and can be used to predict values of the dependent variable from values of the independent variable.

来源:美国大学理事会 AP 课程与考试说明

要为周期数据(潮汐、日照、温度)建模:从模式重复的地方读周期、从最大和最小得到振幅和中线,并设定相移使一个峰落在正确的输入处。技术拟合一个正弦回归。这样一个模型只在它的上下文定义域上可信。

练习卷
3.8

正切函数
大纲
Learning ObjectiveEssential Knowledge

3.8.A
Construct representations of the tangent function using the unit circle.

  • 3.8.A.1 Given an angle of measure $\theta$ in standard position and a unit circle centered at the origin, there is a point, $P$, where the terminal ray intersects the circle. The tangent function, $f(\theta) = \tan\theta$, gives the slope of the terminal ray.
  • 3.8.A.2 Because the slope of the terminal ray is the ratio of the change in the $y$-values to the change in the $x$-values between any two points on the ray, the tangent function is also the ratio of the sine function to the cosine function. Therefore, $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, where $\cos\theta \neq 0$.

3.8.B
Describe key characteristics of the tangent function.

  • 3.8.B.1 Because the slope values of the terminal ray repeat every one-half revolution of the circle, the tangent function has a period of $\pi$.
  • 3.8.B.2 The tangent function demonstrates periodic asymptotic behavior at input values $\theta = \dfrac{\pi}{2} + k\pi$, for integer values of $k$, because $\cos\theta = 0$ at those values.
  • 3.8.B.3 The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes.

来源:美国大学理事会 AP 课程与考试说明

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ 是终边的斜率。它的斜率值每半圈重复,所以它的周期是 $\pi$。它在 $\cos\theta=0$ 的地方(在 $\theta=\tfrac{\pi}{2}+k\pi$)有垂直渐近线,而它在连续的渐近线之间总是递增,在每个零点切换凹凸性。

练习卷
3.9

反三角函数
大纲
Learning ObjectiveEssential Knowledge

3.9.A
Construct analytical and graphical representations of the inverse of the sine, cosine, and tangent functions over a restricted domain.

  • 3.9.A.1 For inverse trigonometric functions, the input and output values are switched from their corresponding trigonometric functions, so the output value of an inverse trigonometric function is often interpreted as an angle measure and the input is a value in the range of the corresponding trigonometric function.
  • 3.9.A.2 The inverse trigonometric functions are called arcsine, arccosine, and arctangent (also represented as $\sin^{-1}x, \cos^{-1}x,$ and $\tan^{-1}x$). Because the corresponding trigonometric functions are periodic, they are only invertible if they have restricted domains.
  • 3.9.A.3 In order to define their respective inverse functions, the domain of the sine function is restricted to $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$, the cosine function to $[0, \pi]$, and the tangent function to $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.

来源:美国大学理事会 AP 课程与考试说明

反三角函数(inverse trigonometric functions)——反正弦(arcsine)、反余弦(arccosine)、反正切(arctangent)——反转三角函数,所以它们取一个比并返回一个角。因为正弦、余弦和正切重复,它们的定义域在一个反函数能存在之前必须被限制(e.g. 正弦到 $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$)。

词汇表 训练
英文 中文 拼音
inverse trigonometric functions 反三角函数 fǎn sān jiǎo hán shù
arcsine 反正弦 fǎn zhèng xián
练习卷
3.10

三角方程与不等式
大纲
Learning ObjectiveEssential Knowledge

3.10.A
Solve equations and inequalities involving trigonometric functions.

  • 3.10.A.1 Inverse trigonometric functions are useful in solving equations and inequalities involving trigonometric functions, but solutions may need to be modified due to domain restrictions.
  • 3.10.A.2 Because trigonometric functions are periodic, there are often infinitely many solutions to trigonometric equations.
  • 3.10.A.3 In trigonometric equations and inequalities arising from a contextual scenario, there is often a domain restriction that can be implied from the context, which limits the number of solutions.

来源:美国大学理事会 AP 课程与考试说明

用反三角函数来解三角方程。因为这些函数是周期性的,通常有无穷多个解——通过加周期的倍数写通解,然后保留在要求的(常常是上下文的)定义域里的那些。

Worked example. 在 $[0,2\pi)$ 上解 $2\sin\theta=1$。那么 $\sin\theta=\tfrac12$,而在单位圆上正弦在第一和第二象限为正,所以 $\theta=\dfrac{\pi}{6}$ 或 $\dfrac{5\pi}{6}$。在所有实数上你会给每个加 $2\pi k$。

练习卷
3.11

正割、余割与余切函数
大纲
Learning ObjectiveEssential Knowledge

3.11.A
Identify key characteristics of functions that involve quotients of the sine and cosine functions.

  • 3.11.A.1 The secant function, $f(\theta) = \sec\theta$, is the reciprocal of the cosine function, where $\cos\theta \neq 0$.
  • 3.11.A.2 The cosecant function, $f(\theta) = \csc\theta$, is the reciprocal of the sine function, where $\sin\theta \neq 0$.
  • 3.11.A.3 The graphs of the secant and cosecant functions have vertical asymptotes where cosine and sine are zero, respectively, and have a range of $(-\infty, -1] \cup [1, \infty)$.
  • 3.11.A.4 The cotangent function, $f(\theta) = \cot\theta$, is the reciprocal of the tangent function, where $\tan\theta \neq 0$. Equivalently, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$, where $\sin\theta \neq 0$.
  • 3.11.A.5 The graph of the cotangent function has vertical asymptotes for domain values where $\tan\theta = 0$ and is decreasing between consecutive asymptotes.

来源:美国大学理事会 AP 课程与考试说明

这些是主要三个的倒数(reciprocals):

$$\sec\theta=\frac{1}{\cos\theta},\qquad \csc\theta=\frac{1}{\sin\theta},\qquad \cot\theta=\frac{1}{\tan\theta}.$$
正割和余割在余弦和正弦为零的地方有垂直渐近线;余切在正切为零的地方(正弦为零的地方)有它们。

sec 是 cos 的倒数,在 cos 为零的地方有渐近线
sec 是 cos 的倒数,在 cos 为零的地方有渐近线
词汇表 训练
英文 中文 拼音
reciprocals 倒数 dào shǔ
练习卷
3.12

三角函数的等价表示
大纲
Learning ObjectiveEssential Knowledge

3.12.A
Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity.

  • 3.12.A.1 The Pythagorean Theorem can be applied to right triangles with points on the unit circle at coordinates $(\cos\theta, \sin\theta)$, resulting in the Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.
  • 3.12.A.2 The Pythagorean identity can be algebraically manipulated into other forms involving trigonometric functions, such as $\tan^2\theta = \sec^2\theta - 1$, and can be used to establish other trigonometric relationships, such as $\arcsin x = \arccos\left(\sqrt{1 - x^2}\right)$, with appropriate domain restrictions.

3.12.B
Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities.

  • 3.12.B.1 The sum identity for sine is $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$.
  • 3.12.B.2 The sum identity for cosine is $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$.
  • 3.12.B.3 The sum identities for sine and cosine can also be used as difference and double-angle identities.
  • 3.12.B.4 Properties of trigonometric functions, known trigonometric identities, and other algebraic properties can be used to verify additional trigonometric identities.

3.12.C
Solve equations using equivalent analytic representations of trigonometric functions.

  • 3.12.C.1 A specific equivalent form involving trigonometric expressions can make information more accessible.
  • 3.12.C.2 Equivalent trigonometric forms may be useful in solving trigonometric equations and inequalities.

来源:美国大学理事会 AP 课程与考试说明

毕达哥拉斯恒等式(Pythagorean identity)来自单位圆上的 $x^2+y^2=1$:

$$\sin^2\theta+\cos^2\theta=1,$$
它重新排列成带 $\sec$ 和 $\tan$ 的形式。和角公式(sum identities):
$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta,\qquad \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta,$$
也给出差角和二倍角(double-angle)恒等式(设 $\beta=\alpha$)。

词汇表 训练
英文 中文 拼音
Pythagorean identity 毕达哥拉斯恒等式 bì dá gē lā sī héng děng shì
sum identities 和角公式 hé jiǎo gōng shì
double-angle 二倍角 èr bèi jiǎo
练习卷
3.13

三角学与极坐标
大纲
Learning ObjectiveEssential Knowledge

3.13.A
Determine the location of a point in the plane using both rectangular and polar coordinates.

  • 3.13.A.1 The polar coordinate system is based on a grid of circles centered at the origin and on lines through the origin. Polar coordinates are defined as an ordered pair, $(r, \theta)$, such that $|r|$ represents the radius of the circle on which the point lies, and $\theta$ represents the measure of an angle in standard position whose terminal ray includes the point. In the polar coordinate system, the same point can be represented many ways.
  • 3.13.A.2 The coordinates of a point in the polar coordinate system, $(r, \theta)$, can be converted to coordinates in the rectangular coordinate system, $(x, y)$, using $x = r\cos\theta$ and $y = r\sin\theta$.
  • 3.13.A.3 The coordinates of a point in the rectangular coordinate system, $(x, y)$, can be converted to coordinates in the polar coordinate system, $(r, \theta)$, using $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\left(\dfrac{y}{x}\right)$ for $x > 0$ or $\theta = \arctan\left(\dfrac{y}{x}\right) + \pi$ for $x < 0$.
  • 3.13.A.4 A complex number can be understood as a point in the complex plane and can be determined by its corresponding rectangular or polar coordinates. When the complex number has the rectangular coordinates $(a, b)$, it can be expressed as $a + bi$. When the complex number has polar coordinates $(r, \theta)$, it can be expressed as $(r\cos\theta) + i(r\sin\theta)$.

来源:美国大学理事会 AP 课程与考试说明

极坐标(polar coordinates)由 $(r,\theta)$ 定位一个点——在角 $\theta$ 距原点的距离 $r$。用以下转换

$$x=r\cos\theta,\quad y=r\sin\theta,\qquad r=\sqrt{x^2+y^2},\quad \tan\theta=\frac{y}{x}.$$
一个复数(complex number)是点 $(x,y)=x+yi$,同样能用 $r$ 和 $\theta$ 写。

极坐标由一个点的距离 r 和角 theta 给出它
极坐标由一个点的距离 r 和角 theta 给出它
词汇表 训练
英文 中文 拼音
Polar coordinates 极坐标 jí zuò biāo
complex number 复数 fù shù
练习卷
3.14

极坐标函数图像
大纲
Learning ObjectiveEssential Knowledge

3.14.A
Construct graphs of polar functions.

  • 3.14.A.1 The graph of the function $r = f(\theta)$ in polar coordinates consists of input-output pairs of values where the input values are angle measures and the output values are radii.
  • 3.14.A.2 The domain of the polar function $r = f(\theta)$, given graphically, can be restricted to a desired portion of the function by selecting endpoints corresponding to the desired angle and radius.
  • 3.14.A.3 When graphing polar functions in the form of $r = f(\theta)$, changes in input values correspond to changes in angle measure from the positive $x$-axis, and changes in output values correspond to changes in distance from the origin.

来源:美国大学理事会 AP 课程与考试说明

极坐标曲线的描绘

$r=f(\theta)$ 的图象是随着 $\theta$ 扫过所有的点 $(r,\theta)$。限制 $\theta$ 的定义域只画曲线的一部分。随着 $\theta$ 增加,$r$ 增长或缩小,描出螺线、圆、玫瑰线和蚶线。

一条心形线 r = 1 + cos theta,直接以极坐标形式草绘
一条心形线 r = 1 + cos theta,直接以极坐标形式草绘
探索

Plot a curve in polar coordinates

In polar form a point is a distance $r$ at angle $\theta$. Letting $r$ depend on $\theta$ traces shapes a rule in $x,y$ can't — like this cardioid.

练习卷
3.15

极坐标函数的变化率
大纲
Learning ObjectiveEssential Knowledge

3.15.A
Describe characteristics of the graph of a polar function.

  • 3.15.A.1 If a polar function, $r = f(\theta)$, is positive and increasing or negative and decreasing, then the distance between $f(\theta)$ and the origin is increasing.
  • 3.15.A.2 If a polar function, $r = f(\theta)$, is positive and decreasing or negative and increasing, then the distance between $f(\theta)$ and the origin is decreasing.
  • 3.15.A.3 For a polar function, $r = f(\theta)$, if the function changes from increasing to decreasing or decreasing to increasing on an interval, then the function has a relative extremum on the interval corresponding to a point relatively closest to or farthest from the origin.
  • 3.15.A.4 The average rate of change of $r$ with respect to $\theta$ over an interval of $\theta$ is the ratio of the change in the radius values to the change in $\theta$ over an interval of $\theta$. Graphically, the average rate of change indicates the rate at which the radius is changing per radian.
  • 3.15.A.5 The average rate of change of $r$ with respect to $\theta$ over an interval of $\theta$ can be used to estimate values of the function within the interval.

来源:美国大学理事会 AP 课程与考试说明

随着 $\theta$ 增加,离原点的距离 $r$ 变化:

  • $r$ 为正且增加(或为负且减少)$\Rightarrow$ 远离原点移动;
  • $r$ 为正且减少(或为负且增加)$\Rightarrow$ 朝向原点移动。

$r$ 在递增和递减之间切换的地方,距离达到一个相对极值。$r$ 相对于 $\theta$ 的平均变化率,$\dfrac{\Delta r}{\Delta \theta}$,估计曲线在一个区间上向内或向外移动多快。

练习卷
3.15

考试技巧
  • 用弧度和单位圆:$\cos\theta$ 和 $\sin\theta$ 是坐标,总是在 $-1$ 和 $1$ 之间。
  • 对于一个正弦型函数 $a\sin(b(\theta+c))+d$:$|a|$ 是振幅、$\tfrac{2\pi}{|b|}$ 是周期、$d$ 是中线、相移是 $-c$($+c$ 向左移)。
  • 通过从最大和最小读周期、振幅和中线为周期数据建模。
  • 反三角函数需要一个限制的定义域并返回一个角;三角方程有无穷多个解(加周期)。
  • 用 $x=r\cos\theta$、$y=r\sin\theta$ 转换极↔直角。

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