Trigonometry and polar coordinates: synthesis and challenges

Summary

This lesson closes the unit on trigonometric functions and polar coordinates. It gathers into a single deductive chain what was proved in its six lessons: the measure of angles by arc length, the definition of the sine and the cosine on the unit circle and its consequences, the study of these functions and of their inverses, the solution of all triangles, and polar coordinates with their curves. It explains how that chain rests on the course Algebra and Geometry I, from division with remainder to conics and vectors, and how it prepares the analytic trigonometry of the following unit. In an exit assessment it repeats the skills of the entry assessment of the introduction, adds two questions about the reasons behind the results, and ends with ten challenges, harder than the problems of the lessons, which combine several of them and each demand an idea of one’s own.

Learning objectives

Reconstruct the chain of definitions and theorems of the unit, from the radian to the polar equation of conics, identifying the result on which each one depends.
Relate the results of the unit to those of Course I and to the problems that analytic trigonometry leaves open.
Solve problems that combine the measure of arcs, the symmetries and the graphs of the trigonometric functions, their inverses, the solution of triangles and polar coordinates.

What the unit achieved

From angles to functions

The unit starts from a few admitted facts, which are worth listing exactly. The first two require infinitesimal calculus: the facts about lengths and areas (every arc has a length and every region bounded by line segments and arcs has an area; both are additive, do not change when the figures are moved and grow with them; the arcs subtended by the same central angle are proportional to the radii, and the disk of radius $r$ has area $\pi r^2$ ), and the axiom of traversal by arc length: for each real number $s$ there is a unique point $P(s)$ of the unit circle that is reached by traveling along it from $(1, 0)$ a length $\abs{s}$ , in the direction indicated by the sign of $s$ ; that traversal can be concatenated (advancing $t$ from $P(s)$ leads to $P(s + t)$ ), it has period $2\pi$ , it reaches every point and, in the first half turn, which ends at $P(\pi) = (-1, 0)$ , it runs over the upper semicircle with strictly decreasing abscissa. The others come from elementary geometry, which the course does not develop: the sum $\pi$ of the angles of a triangle, the equality of the base angles of an isosceles triangle, the sum of adjacent angles and the position of the center of a circle relative to an inscribed angle, the similarity criteria, the Pythagorean theorem, the preservation of lengths by reflections, which reverse the direction of rotation, and by the half-turn rotation, which preserves it, and the comparison of the angles under which a chord is seen from points inside and outside the disk. Along with them, the value of $\pi$ is also admitted, whose decimal expansion begins with 3.14159, so that $3 < \pi < 4$ , as well as its irrationality. Two other facts, which also require infinitesimal calculus, are admitted in the lesson on the trigonometric functions and their graphs, and only some problems use them, never the theorems of the chain: the continuity of the trigonometric functions and of their inverses, thanks to which the intermediate value theorem applies to them, which is not admitted anew but cited from Course I, and the strict concavity of the sine on each interval $\icc{2k\pi}{(2k + 1)\pi}$ ; and the last solved problem of the lesson on trigonometric ratios admits, for the same reason, that the areas of the inscribed regular polygons come as close as desired to that of the disk. On that basis, the lesson on angles and their measure defined the number $\pi$ as half the length of the unit circle and radian measure as the signed length of the arc traveled; by reflecting the first half turn in the $x$ -axis, it proved that in the second one the traversal runs over the lower semicircle with strictly increasing abscissa; from that same basis followed the conversion between degrees and radians, the formulas $s = r\theta$ and $A = \frac{1}{2} r^2 \theta$ , the inscribed angle theorem, by which such an angle measures half the arc it intercepts, and the theorem of reduction to the first turn, according to which every real number can be written in a unique way as $\theta_0 + 2k\pi$ with $\theta_0$ in $\ico{0}{2\pi}$ . Since the axiom implies that one turn does not pass twice through the same point, two numbers determine the same point $P(\theta)$ exactly when they differ by an integer multiple of $2\pi$ .

The lesson on trigonometric ratios and the unit circle gave two definitions of the sine, the cosine and the tangent: as quotients of the sides of a right triangle, which similarity makes independent of the triangle, and as the coordinates of $P(\theta)$ , valid for every real number; it proved that they agree for acute angles. From the second definition were deduced the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ , the special values of $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ , and the symmetries of the circle, which give parity, periodicity and the relations with supplementary and complementary angles. From this follows the reference angle method, which reduces the computation of the ratios of any angle to that of an acute angle.

In the lesson on the trigonometric functions and their graphs, those ratios became functions of a real variable. The lemma on periods, proved with the Archimedean property, and the location of the maxima showed that $2\pi$ is the fundamental period of the sine and the cosine, and the location of the zeros, that $\pi$ is that of the tangent; the graphs were built by unrolling the circle, and the comparison of the areas of two triangles and a sector gave the inequality $\sin x < x < \tan x$ for $0 < x < \frac{\pi}{2}$ , from which follow the bound $\abs{\sin x} \leq \abs{x}$ and the vertical asymptotes of the tangent. The sinusoids $a \sin(b(x - c)) + d$ , obtained from the sine by the transformations of the Course I unit on algebraic functions, model every periodic phenomenon of simple form.

Inversion and the measurement of triangles

Now, a periodic function is not injective. The lesson on the inverse trigonometric functions read in the traversal axiom that the cosine is strictly decreasing on $\icc{0}{\pi}$ , because that is the half-turn clause written in coordinates; from this and from the symmetry of complementary angles it proved that the sine is strictly increasing on $\icc{-\frac{\pi}{2}}{\frac{\pi}{2}}$ , and from both, that the tangent is increasing on $\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}$ ; it defined the arcsine, the arccosine and the arctangent on those intervals, and distinguished the equality $\sin(\arcsin x) = x$ , always true, from $\arcsin(\sin x) = x$ , true only on the principal interval. With the symmetries, the inverses solved the elementary equations over a full turn.

With this, everything was ready for the lesson on solving triangles. When one vertex is placed at the origin, the vertex opposite a side has coordinates $\left( b \cos A, b \sin A \right)$ ; from there follow, in this order, the area $\frac{1}{2} bc \sin A$ , the law of sines and, with the inscribed angle theorem, its relation to the radius of the circumscribed circle, the law of cosines, which extends the Pythagorean theorem, and Heron’s formula. Since the cosine determines an angle of a triangle and the sine does not, the case of two sides and the angle opposite one of them admits no solution, one or two, and the ambiguous case theorem counts them through the roots of a quadratic equation. Finally, the law of cosines and the expansion of the square of the norm of a difference gave the measure of the angle between two vectors through the dot product.

A second coordinate system

The lesson on polar coordinates reversed the point of view: instead of computing the Cartesian coordinates of a point from its distance to the origin and its angle, it named the point by those two numbers. Admitting negative radii, it described all the representations of a point, proved the conversion formulas, with the arctangent corrected according to the quadrant, and obtained from the law of cosines the distance between two points in polar coordinates. The symmetry tests and the tables of values made it possible to sketch cardioids, roses, spirals and limaçons, and the focus-directrix theorem was transformed into the equation $r = \frac{ed}{1 + e \cos \theta}$ , in which the type, the vertices and the latus rectum of each conic can be read.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that come from Course I or are admitted without proof, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

ATraversal by arc length. For each real number $s$ there is a unique point $P(s)$ of the unit circle that is reached by traveling along it from $(1, 0)$ a length $\abs{s}$ , counterclockwise if $s > 0$ and clockwise if $s < 0$ . Advancing from $P(s)$ a length $t$ counterclockwise leads to $P(s + t)$ , and clockwise, to $P(s - t)$ ; we have $P(s + 2\pi) = P(s)$ ; every point of the circle is $P(s)$ for some $s$ ; $P(\pi) = (-1, 0)$ , and the abscissa of $P(s)$ strictly decreases, with nonnegative ordinate, as $s$ increases from $0$ to $\pi$ . It is admitted without proof, because it rests on arc length and on the intermediate value property; the second half turn is deduced from the first with reflections. Admitted without proof in this unit.
BLengths and areas. Every arc has a length and every region bounded by line segments and arcs has an area; congruent figures have the same measure, measures are additive, the area of a region that contains another is not smaller, and it is larger if the difference contains a disk; the arcs subtended by the same central angle are proportional to the radii, and the disk of radius $r$ has area $\pi r^2$ . It is admitted without proof, because lengths and curved areas are defined by means of limits. Admitted without proof in this unit.
CReflections and rotations. A reflection in a line preserves distances and arc lengths, and reverses the direction of rotation; the half-turn rotation about the origin preserves them and preserves the direction. It is admitted from elementary geometry. Admitted without proof in this unit.
DElementary geometry. The three angles of a triangle add up to $\pi$ ; the base angles of an isosceles triangle are equal; a ray that passes through the interior of an angle divides it into two that add up to the given angle; if the center lies inside an inscribed angle, the diameter starting from the vertex divides it in two and cuts the intercepted arc, and if it lies outside, one of the sides lies between that diameter and the other side; the similarity criteria for triangles and the Pythagorean theorem; and a point outside the disk sees a chord under an angle smaller than the inscribed angle on the same side, and a point inside, under a larger angle. They are admitted from elementary geometry. Admitted without proof in this unit.
EArchimedean property. Given two positive real numbers $p$ and $q$ , some natural multiple of $q$ exceeds $p$ ; from it follow the density of the rational numbers and the existence of the least integer greater than a given real number. Proved in Course I, in the unit on number systems. Admitted without proof in this unit.
FFunctions and polynomials. A strictly monotonic function is injective, the graph of its inverse is the reflection of its own graph in the line $y = x$ , transformations shift and stretch graphs, and sign tables and Vieta’s formulas govern the signs of quotients and the roots of polynomials. Proved in Course I, in the unit on algebraic functions and polynomials. Admitted without proof in this unit.
GAnalytic geometry. The distance formula, the equation of the circle, the slope of a line, vectors with the dot product, and the focus-directrix characterization of conics. Proved in Course I, in the unit on the analytic geometry of the plane. Admitted without proof in this unit.
HSymmetries. The sine is odd and the cosine even; $\sin(\pi - \theta) = \sin \theta$ , $\cos(\pi - \theta) = -\cos \theta$ , $\sin(\pi + \theta) = -\sin \theta$ , $\cos(\pi + \theta) = -\cos \theta$ and $\cos \theta = \sin\left( \frac{\pi}{2} - \theta \right)$ , and both are periodic with period $2\pi$ . Established in: 10. Symmetries and periodicity.

Segment 1 From angles to functions
- Measure angles by the length of the arc they travel, relate the inscribed angle to the arc it intercepts, and reduce every real number to the first turn.
- Define the sine and the cosine on the unit circle and prove that they agree with the ratios of the right triangle.
- Deduce the Pythagorean identity, the special values and the symmetries, and with them the reference angle method.
- Study the sine, the cosine and the tangent as functions: their fundamental periods, their graphs, the inequality $\sin x < x < \tan x$ and the sinusoids.
1. Radian measure. With $\pi$ equal to half the length of the unit circle, the radian measure of an oriented angle is the signed length of the arc it travels; every real number is the measure of an angle, and an angle of $g$ degrees measures $\frac{\pi g}{180}$ radians.
  Rests on: A (Traversal by arc length), B (Lengths and areas). Proved in: Angles and their measure: degrees and radians.
2. Arc and sector. In a circle of radius $r$ , a central angle of $\theta$ radians, with $0 < \theta \leq 2\pi$ , subtends an arc of length $s = r\theta$ and a sector of area $\frac{1}{2} r^2 \theta$ .
  Uses: 1. Radian measure. Rests on: B (Lengths and areas), E (Archimedean property). Proved in: Angles and their measure: degrees and radians.
3. Inscribed angle. An inscribed angle in a circle of radius $r$ that intercepts an arc of length $s$ measures $\frac{s}{2r}$ ; therefore, the inscribed angles on the same chord are equal or supplementary according as their vertices lie on the same arc or on different arcs, and the angle inscribed in a semicircle is a right angle.
  Uses: 2. Arc and sector. Rests on: B (Lengths and areas), D (Elementary geometry). Proved in: Angles and their measure: degrees and radians.
4. Reduction to the first turn. Every real number $\theta$ can be written in a unique way as $\theta_0 + 2k\pi$ , with $k$ an integer and $\theta_0$ in $\ico{0}{2\pi}$ ; since one turn does not repeat points, two angles are coterminal if and only if their difference is an integer multiple of $2\pi$ .
  Uses: 1. Radian measure. Rests on: A (Traversal by arc length), E (Archimedean property), C (Reflections and rotations). Proved in: Angles and their measure: degrees and radians.
5. Ratios of an acute angle. In the right triangles with an acute angle $\alpha$ , the quotients of the sides depend only on $\alpha$ , and the sine and the cosine of $\alpha$ lie strictly between $0$ and $1$ .
  Rests on: D (Elementary geometry). Proved in: Trigonometric ratios and the unit circle.
6. Sine and cosine of a real number. For every real number $\theta$ , the cosine and the sine of $\theta$ are the abscissa and the ordinate of the point $P(\theta)$ of the unit circle, and the tangent is their quotient $\frac{\sin \theta}{\cos \theta}$ when the cosine does not vanish; their signs in each quadrant are deduced from the half-turn property and from its reflection, the second half turn, and those of any angle are those of its reduction to the first turn.
  Uses: 1. Radian measure, 4. Reduction to the first turn. Rests on: A (Traversal by arc length), C (Reflections and rotations). Proved in: Trigonometric ratios and the unit circle.
7. Agreement of the definitions. If $0 < \theta < \frac{\pi}{2}$ , the sine, the cosine and the tangent defined on the unit circle coincide with those defined in a right triangle with an acute angle of measure $\theta$ .
  Uses: 5. Ratios of an acute angle, 6. Sine and cosine of a real number. Rests on: A (Traversal by arc length), C (Reflections and rotations), G (Analytic geometry). Proved in: Trigonometric ratios and the unit circle.
8. Pythagorean identity. For every real number $\theta$ we have $\sin^2 \theta + \cos^2 \theta = 1$ , and from it $1 + \tan^2 \theta = \sec^2 \theta$ when $\cos \theta \neq 0$ .
  Uses: 6. Sine and cosine of a real number. Rests on: G (Analytic geometry). Proved in: Trigonometric ratios and the unit circle.
9. Special values. The sine and the cosine of $\frac{\pi}{4}$ are equal to $\frac{\sqrt{2}}{2}$ ; the sine of $\frac{\pi}{6}$ and the cosine of $\frac{\pi}{3}$ are equal to $\frac{1}{2}$ , and the cosine of $\frac{\pi}{6}$ and the sine of $\frac{\pi}{3}$ are equal to $\frac{\sqrt{3}}{2}$ .
  Uses: 1. Radian measure, 6. Sine and cosine of a real number. Rests on: G (Analytic geometry), D (Elementary geometry), C (Reflections and rotations). Proved in: Trigonometric ratios and the unit circle.
10. Symmetries and periodicity. The sine is odd and the cosine even; an angle and its supplementary angle have the same sine and opposite cosines, $\theta$ and $\pi + \theta$ have opposite sines and cosines, the sine of an angle is the cosine of its complementary angle, and both are periodic with period $2\pi$ ; therefore, the sine and the cosine of an angle are, up to the sign of its quadrant, those of its reference angle.
  Uses: 4. Reduction to the first turn, 6. Sine and cosine of a real number. Rests on: A (Traversal by arc length), C (Reflections and rotations), F (Functions and polynomials). Proved in: Trigonometric ratios and the unit circle.
11. Properties of the sine and the cosine. The sine and the cosine have range $\icc{-1}{1}$ ; the sine vanishes exactly at the numbers $k\pi$ and equals $1$ exactly at the numbers $\frac{\pi}{2} + 2k\pi$ , and the cosine equals $1$ exactly at the numbers $2k\pi$ , with $k$ an integer.
  Uses: 4. Reduction to the first turn, 6. Sine and cosine of a real number, 8. Pythagorean identity. Rests on: A (Traversal by arc length), H (Symmetries). Proved in: The trigonometric functions and their graphs.
12. Fundamental period. If $q$ is the fundamental period of a function, its periods are exactly the numbers $kq$ , with $k$ a natural number; and $2\pi$ is the fundamental period of the sine and the cosine.
  Uses: 11. Properties of the sine and the cosine. Rests on: E (Archimedean property). Proved in: The trigonometric functions and their graphs.
13. Sine, arc and tangent. If $0 < x < \frac{\pi}{2}$ , then $\sin x < x < \tan x$ ; from this it follows that $\abs{\sin x} \leq \abs{x}$ for every real number $x$ , with equality only at $x = 0$ .
  Uses: 2. Arc and sector, 6. Sine and cosine of a real number, 11. Properties of the sine and the cosine. Rests on: B (Lengths and areas), G (Analytic geometry), H (Symmetries). Proved in: The trigonometric functions and their graphs.
14. Properties of the tangent. The tangent is odd, has fundamental period $\pi$ , vanishes exactly at the numbers $k\pi$ , its range is $\mathbb{R}$ , and the lines $x = \frac{\pi}{2} + k\pi$ are vertical asymptotes of its graph.
  Uses: 8. Pythagorean identity, 11. Properties of the sine and the cosine, 13. Sine, arc and tangent. Rests on: A (Traversal by arc length), F (Functions and polynomials), H (Symmetries). Proved in: The trigonometric functions and their graphs.
15. Sinusoids. The sinusoid $y = a \sin(b(x - c)) + d$ , with $a > 0$ and $b > 0$ , is obtained from the sine by transformations; its range is $\icc{d - a}{d + a}$ and its fundamental period is $\frac{2\pi}{b}$ .
  Uses: 11. Properties of the sine and the cosine, 12. Fundamental period. Rests on: F (Functions and polynomials). Proved in: The trigonometric functions and their graphs.
Segment 2 Inversion and the measurement of triangles
- Deduce from the traversal axiom the strict monotonicity of the cosine and the sine, and from it that of the tangent, on their principal intervals, and define their inverse functions.
- Distinguish the equality $\sin(\arcsin x) = x$ , always true, from $\arcsin(\sin x) = x$ , and solve the elementary equations over a full turn.
- Deduce from the coordinates of a vertex the area of the triangle, the law of sines, the law of cosines and Heron’s formula.
- Count the solutions of the ambiguous case and measure the angle between two vectors with the dot product.
1. Monotonicity and bijective restrictions. By the half-turn property of the traversal axiom, the cosine is strictly decreasing on $\icc{0}{\pi}$ ; from this and from the symmetries, the sine is strictly increasing on $\icc{-\frac{\pi}{2}}{\frac{\pi}{2}}$ and the tangent on $\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}}$ ; restricted to those intervals, they are bijections onto $\icc{-1}{1}$ , $\icc{-1}{1}$ and $\mathbb{R}$ .
  Uses: 4. Reduction to the first turn, 6. Sine and cosine of a real number, 14. Properties of the tangent. Rests on: A (Traversal by arc length), F (Functions and polynomials), H (Symmetries). Proved in: The inverse trigonometric functions.
2. Inverse functions. The arcsine, the arccosine and the arctangent, inverses of those restrictions, satisfy $\sin(\arcsin x) = x$ on $\icc{-1}{1}$ , whereas $\arcsin(\sin x) = x$ only on $\icc{-\frac{\pi}{2}}{\frac{\pi}{2}}$ ; moreover, $\arcsin x + \arccos x = \frac{\pi}{2}$ and $\cos(\arcsin x) = \sqrt{1 - x^2}$ .
  Uses: 8. Pythagorean identity, 16. Monotonicity and bijective restrictions. Rests on: F (Functions and polynomials), H (Symmetries). Proved in: The inverse trigonometric functions.
3. Elementary equations. If $-1 < c < 1$ and $\alpha = \arcsin c$ , the equation $\sin x = c$ has exactly two solutions in $\ico{0}{2\pi}$ , namely $\pi - \alpha$ and, according to the sign of $\alpha$ , either $\alpha$ itself or $2\pi + \alpha$ ; the equation $\cos x = c$ has there the solutions $\arccos c$ and $2\pi - \arccos c$ .
  Uses: 16. Monotonicity and bijective restrictions, 17. Inverse functions. Rests on: H (Symmetries). Proved in: The inverse trigonometric functions.
4. Area of the triangle. With the vertex $A$ at the origin and the vertex $B$ on the positive half of the horizontal axis, the vertex $C$ has coordinates $\left( b \cos A, b \sin A \right)$ , and the area of the triangle is $\frac{1}{2} bc \sin A$ .
  Uses: 6. Sine and cosine of a real number. Rests on: G (Analytic geometry), H (Symmetries). Proved in: Solving triangles: the laws of sines and cosines.
5. Law of sines. The sides of a triangle are proportional to the sines of the opposite angles, and the common quotient $\frac{a}{\sin A}$ is the diameter $2R$ of the circumscribed circle.
  Uses: 7. Agreement of the definitions, 3. Inscribed angle, 19. Area of the triangle. Rests on: D (Elementary geometry), H (Symmetries). Proved in: Solving triangles: the laws of sines and cosines.
6. Law of cosines. In every triangle we have $a^2 = b^2 + c^2 - 2bc \cos A$ , which reduces to the Pythagorean theorem when the angle $A$ is a right angle.
  Uses: 8. Pythagorean identity, 19. Area of the triangle. Rests on: G (Analytic geometry). Proved in: Solving triangles: the laws of sines and cosines.
7. Heron’s formula. The area of a triangle with sides $a$ , $b$ and $c$ and semiperimeter $s$ is $\sqrt{s(s - a)(s - b)(s - c)}$ .
  Uses: 8. Pythagorean identity, 19. Area of the triangle, 21. Law of cosines. Rests on: F (Functions and polynomials). Proved in: Solving triangles: the laws of sines and cosines.
8. Ambiguous case. Given an angle $A$ , the adjacent side $c$ and the opposite side $a$ , the possible triangles correspond to the positive roots of $b^2 - 2bc \cos A + c^2 - a^2 = 0$ , and there are none, one or two depending on how $a$ compares with $c \sin A$ and with $c$ .
  Uses: 6. Sine and cosine of a real number, 8. Pythagorean identity, 21. Law of cosines. Rests on: F (Functions and polynomials). Proved in: Solving triangles: the laws of sines and cosines.
9. Angle between vectors. If $\theta$ is the angle between the nonzero vectors $\vec{u}$ and $\vec{v}$ , then $\cos \theta = \frac{\vec{u} \cdot \vec{v}}{\norma{\vec{u}} \norma{\vec{v}}}$ .
  Uses: 21. Law of cosines. Rests on: G (Analytic geometry). Proved in: Solving triangles: the laws of sines and cosines.
Segment 3 A second coordinate system
- Describe all the polar representations of a point, admitting negative radii.
- Convert Cartesian coordinates to polar coordinates with the arctangent corrected according to the quadrant.
- Compute distances in polar coordinates and sketch polar curves with the symmetry tests.
- Write the conics with a focus at the pole and read in their equation the type, the vertices and the latus rectum.
1. Representations of a point. The pairs $(r_1, \theta_1)$ and $(r_2, \theta_2)$ represent the same point other than the pole if and only if $r_2 = r_1$ and $\theta_2 - \theta_1$ is an even multiple of $\pi$ , or else $r_2 = -r_1$ and $\theta_2 - \theta_1$ is an odd multiple of $\pi$ .
  Uses: 4. Reduction to the first turn, 6. Sine and cosine of a real number, 8. Pythagorean identity. Rests on: H (Symmetries). Proved in: Polar coordinates and polar curves.
2. From Cartesian to polar. Every point $(x, y)$ other than the pole has a unique representation with $r = \sqrt{x^2 + y^2}$ and $\theta$ in $\ioc{-\pi}{\pi}$ ; if $x \neq 0$ , its angle is $\arctan \frac{y}{x}$ , to which $\pi$ is added or from which it is subtracted when $x < 0$ .
  Uses: 25. Representations of a point, 17. Inverse functions. Rests on: A (Traversal by arc length), G (Analytic geometry), H (Symmetries). Proved in: Polar coordinates and polar curves.
3. Distance in polar coordinates. The distance $d$ between the points $(r_1, \theta_1)$ and $(r_2, \theta_2)$ satisfies $d^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_2 - \theta_1)$ , whatever the signs of the radii.
  Uses: 4. Reduction to the first turn, 21. Law of cosines, 25. Representations of a point. Rests on: H (Symmetries). Proved in: Polar coordinates and polar curves.
4. Symmetry tests. If the equation of a polar curve becomes an equivalent one when $\theta$ is replaced by $-\theta$ , $\theta$ by $\pi - \theta$ or $r$ by $-r$ , the curve is symmetric, respectively, with respect to the polar axis, the line $\theta = \frac{\pi}{2}$ or the pole.
  Uses: 6. Sine and cosine of a real number. Rests on: H (Symmetries). Proved in: Polar coordinates and polar curves.
5. Polar equation of conics. If $e > 0$ and $d > 0$ , the points whose distance to the pole is $e$ times their distance to the line $x = d$ form the graph of $r = \frac{ed}{1 + e \cos \theta}$ .
  Uses: 25. Representations of a point. Rests on: G (Analytic geometry), H (Symmetries). Proved in: Polar coordinates and polar curves.
6. Elements of the polar conic. In the conic $r = \frac{ed}{1 + e \cos \theta}$ , the polar axis is the focal axis, one vertex is at distance $\frac{ed}{1 + e}$ from the pole and the latus rectum measures $2ed$ ; if $e < 1$ , the semi-major axis is $\frac{ed}{1 - e^2}$ .
  Uses: 28. Symmetry tests, 29. Polar equation of conics. Rests on: G (Analytic geometry). Proved in: Polar coordinates and polar curves.

Toward analytic trigonometry. The chain stops before the sine and the cosine of a sum of angles: the symmetries give them only when one of the summands is a multiple of $\frac{\pi}{2}$ . The following unit deduces the sum and difference formulas, and with them proves identities, solves trigonometric equations in general and classifies second-degree equations by means of the rotation of axes.

The unit within the course

The unit, the first of Course II, rests on Course I at precise points. Reduction to the first turn is the division with remainder of the Course I lesson on divisibility, carried over to the real numbers with the Archimedean property of the Course I lesson on the decimal expansion of rational numbers, and the area of the sector was extended to all angles with the density of the rational numbers from the Course I lesson on the real line. The inverses applied the theorems on the inverse function and on transformations from the Course I lesson on function operations, and the asymptotes were studied with sign tables. The unit circle, slopes, vectors, the dot product and the conics come from the Course I unit on the analytic geometry of the plane, whose lesson on conic sections gave the focus-directrix theorem. And the Course I lesson on the hyperbolic functions anticipated the analogy that this unit completes: the sine and the cosine describe the circle $x^2 + y^2 = 1$ just as the hyperbolic sine and cosine describe the hyperbola $x^2 - y^2 = 1$ .

There remains, however, an unanswered question that runs through the whole unit: how are the sine and the cosine of a sum of angles computed from those of each summand? The symmetries settle it only when one of the summands is a multiple of $\frac{\pi}{2}$ ; the values at $\frac{\pi}{12}$ or at $\frac{\pi}{10}$ require special geometric constructions; and the product of two complex numbers in polar form, announced in the lesson on polar coordinates, depends on it. The following unit, which begins with the introduction to analytic trigonometry, answers that question with the sum and difference formulas, and with them proves identities, solves trigonometric equations in general and classifies second-degree equations by means of the rotation of axes.