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Trigonometric functions and polar coordinates: introduction

Summary

This lesson opens the course Algebra and Geometry II and its first unit, devoted to the trigonometric functions and polar coordinates. It offers an entry assessment, which measures the starting point in the six lessons of the unit and in what the unit takes as known from Course I. It gathers what Course I leaves established and presents the six lessons of the unit in the order in which they are studied, from the measure of angles to polar curves and the conics with a focus at the pole, and explains what each one proves and why it matters. It then reconstructs the historical background: Greek astronomy, which needed to measure the inaccessible and tabulated the chords of arcs; the half of the chord of the double arc, which the astronomers of India placed at the center of computation and which, through the Arabic and Latin translations, received the name of sine; trigonometry as an independent discipline, and its transformation, in the eighteenth century, into the study of functions of a real variable. It states the problem that gave rise to it, that of calculating distances from angles, with the triangle of Aristarchus of Samos, and ends by pointing out where these ideas are used in the rest of the course, in other courses on the site and in science and technology.

Learning objectives

  1. Describe the content of the unit, placing each of its lessons in a sequence that runs from the measure of angles to polar curves.
  2. Explain the astronomical and geodetic origin of trigonometry and of polar coordinates, and recognize their main stages, from the table of chords to the functions of a real variable.
  3. Express the chord of an arc by means of the sine of the half arc, and set up the problem of Aristarchus as the calculation of a quotient of distances from an angle.
  4. Identify the places in the course, in other courses and in science and technology where the trigonometric functions and polar coordinates are used.

Entry assessment

Before working through the unit, it is advisable to measure the starting point, without any grading. The first ten questions cover the six content lessons of the unit: two for the inverse functions and two for solving triangles, which are the lessons on which the rest of the course relies most; two for the trigonometric functions and their graphs, one of them a problem in context that has to be set up; two for polar coordinates, one of them on the polar equation of the conics; and one for each of the other two lessons. The same skills are measured again at the end, in the synthesis lesson. The last six check what the course takes as known from Course I: the completeness of the real numbers, the inverse of a function, the tangent line to a circle and the dot product, the eccentricity of an ellipse, the exponential and the logarithm by way of the hyperbolic cosine, and the quotient of two complex numbers in rectangular form. Whoever misses one of them will know which lesson of Course I is worth returning to before starting, and whoever misses one of the first ten, in which lesson of the unit to dwell.

Entry assessment

Before starting the unit, measure your starting point. It is a short self-assessment: it is not graded and its result is stored only in your browser.

  1. Question 1In a circle of radius 99, how long is the arc intercepted by a central angle of 40∘40^\circ?

  2. Question 2If sin⁡θ=35\sin \theta = \frac{3}{5} and θ\theta lies in the second quadrant, calculate tan⁡θ\tan \theta.

  3. Question 3What is the fundamental period of the function f(x)=3sin⁡(4x−π)+1f(x) = 3 \sin(4x - \pi) + 1?

  4. Question 4A mass hanging from a spring oscillates vertically. Its height above the floor varies as a sinusoid of time, between 2020 and 6060 centimeters, with a period of 44 seconds, and it reaches its highest point at the instant t=1t = 1. How many seconds of each oscillation does it spend more than 5050 centimeters above the floor?

  5. Question 5What is the value of arccos⁡(−32)\arccos\left( -\frac{\sqrt{3}}{2} \right)?

  6. Question 6What is the value of arcsin⁡(sin⁡5π7)\arcsin\left( \sin \frac{5\pi}{7} \right)?

  7. Question 7In a triangle, b=5b = 5, c=8c = 8 and the included angle measures A=60∘A = 60^\circ. Calculate the side aa.

  8. Question 8How many triangles are there with A=30∘A = 30^\circ, c=10c = 10 and a=6a = 6?

  9. Question 9What are the polar coordinates, with r>0r > 0 and θ\theta in (−π,π]\ioc{-\pi}{\pi}, of the point with Cartesian coordinates (−1,3)\left( -1, \sqrt{3} \right)?

  10. Question 10The curve with polar equation r=215+2cos⁡θr = \frac{21}{5 + 2 \cos \theta} has a focus at the pole. What curve is it?

  11. Question 11From earlier unitsWhat is the supremum of the set A={1−1n:n∈N}A = \left\{ 1 - \frac{1}{n} : n \in \mathbb{N} \right\}?

  12. Question 12From earlier unitsWhich of these restrictions of the function f(x)=x2f(x) = x^2 has an inverse function?

  13. Question 13From earlier unitsWhat is the equation of the tangent line to the circle x2+y2=25x^2 + y^2 = 25 at its point (3,4)(3, 4)?

  14. Question 14From earlier unitsWhat is the eccentricity of the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1?

  15. Question 15From earlier unitsIf t>0t > 0 and cosh⁡t=54\cosh t = \frac{5}{4}, what is the value of tt?

  16. Question 16From earlier unitsWhat is the value of the quotient 5+5i1+2i\frac{5 + 5i}{1 + 2i}?

The contents of the unit

With this unit begins the course Algebra and Geometry II, which assumes that the course Algebra and Geometry I has been completed. From that course it brings five things that are used here without being proved again. The first is the real numbers as a complete ordered field, with the Archimedean property and the density of the rationals, which will make it possible to reduce every angle to the first turn and to extend to all angles what is proved for the rational ones. The second is the study of functions: domain, range, composition, inverse of a bijective function and graph transformations, with sign charts and the factorization of polynomials. The third is analytic geometry of the plane: distance, the line in Cartesian and parametric form, vectors with the dot product, the circle and the conics with their focus and their directrix. The fourth is the exponential, logarithmic and hyperbolic functions, whose parametrization of the hyperbola anticipates that of the circle. The fifth is the complex numbers in rectangular form, represented as points of the plane, which this unit prepares to be written with a modulus and an angle.

Now, none of those tools answers a very old question: given an angle, what quotient do the sides of a right triangle containing it keep between them? This unit answers it, and from the answer arise six functions of a real variable, their inverses, the solving of all triangles and a second way of naming the points of the plane. The lessons follow one another in this order.

  1. Angles and their measure defines oriented angles, which measure rotations together with their direction and may exceed a full turn, states the axiom of traversal of the circle by arc length, on which the whole unit rests, and uses it to define radian measure as the length of the arc of the unit circle; it proves the formulas for the arc and the sector, the inscribed angle theorem and the reduction of every angle to the first turn, which is the division with remainder of the real numbers by 2π2\pi. It matters because the radian is the measure that makes all the later formulas simple.
  2. Trigonometric ratios and the unit circle defines the sine, the cosine and the tangent first in the right triangle, proving by similarity that they do not depend on the triangle, and then for every real number, as coordinates of a point of the unit circle; it proves the Pythagorean identity, the special values and the symmetries. It is the decisive move of the unit: it turns a geometric ratio into a quantity defined for any angle.
  3. The trigonometric functions and their graphs studies the sine, the cosine and the tangent as functions, proves their fundamental periods, constructs their graphs and proves, by comparing areas, the inequality sin⁡x<x<tan⁡x\sin x < x < \tan x. With the sinusoids it provides the model of all periodic phenomena.
  4. The inverse trigonometric functions deduces from the traversal axiom that the cosine and the sine are strictly monotonic on certain intervals, restricts the functions to them, defines the arcsine, the arccosine and the arctangent, and solves the elementary equations in one turn. It makes it possible to travel the road in reverse: from a known ratio to the angle that produces it.
  5. Solving triangles proves the area in terms of two sides and the included angle, the law of sines, the law of cosines and Heron’s formula, and discusses the ambiguous case. It extends the trigonometry of the right triangle to all triangles and measures, with the dot product, the angle between two vectors.
  6. Polar coordinates and polar curves names each point by its distance to the pole and by an angle, derives the conversion formulas to Cartesian coordinates, sketches cardioids, roses, spirals and limaçons, and obtains the polar equation of the conics with a focus at the pole. It prepares the polar form of complex numbers and the description of orbits.

The unit ends with the synthesis and challenges lesson, which gathers what has been proved, measures with an exit assessment what has been achieved relative to the entry assessment of this lesson, and proposes ten problems of greater difficulty.

Historical background

Trigonometry was not born as a chapter of algebra, but as a technique of astronomy. To predict the position of the heavenly bodies and to measure distances that cannot be traveled, astronomers needed to pass from angles, which are observed, to lengths, which are calculated; and Greek geometry, which knew how to construct figures with straightedge and compass, offered no means whatever of calculating, given an arbitrary angle, the ratio between two sides of a triangle.

The table of chords

Greek astronomy inherited from Babylonian astronomy the division of the circle into 360360 degrees and calculation in base sixty. Around the third century before the Common Era, Aristarchus of Samos estimated with geometric arguments the ratio between the distances of the Sun and of the Moon, and Eratosthenes measured the circumference of the Earth from an angle, as will be seen in the lesson on angles and their measure. To Hipparchus of Nicaea, in the second century before the Common Era, is attributed the first table of chords: for each central angle, the length of the chord it subtends in a circle of fixed radius. The table that has come down to us is that of Claudius Ptolemy, who in the Almagest, in the second century of the Common Era, gave the chords of the arcs at intervals of half a degree in a circle of radius 6060. Menelaus of Alexandria, around the year 100, had extended the calculation to triangles drawn on the celestial sphere.

The chord and the sine are the same information written in two ways. The proposition below makes this precise, and shows why the passage from the one to the other was something more than a change of name: the sine is half the chord of the double arc, and it is a leg of a right triangle.

PropositionThe chord and the sine

In the circle of radius 11 and center OO, let C(θ)C(\theta) be the length of the chord joining the endpoints AA and BB of an arc with central angle θ\theta, with 0<θ<π0 < \theta < \pi. Then C(θ)=2sin⁡θ2C(\theta) = 2 \sin \frac{\theta}{2}.

Proof

  1. C(θ)=d⁡(A,B),d⁡(O,A)=d⁡(O,B)=1C(\theta) = \operatorname{d}(A, B), \quad \operatorname{d}(O, A) = \operatorname{d}(O, B) = 1

    The chord is the side opposite the angle θ\theta in the triangle with vertices OO, AA and BB, whose other two sides are radii.

  2. d⁡(A,M)=d⁡(M,B),∠AOM=∠MOB=θ2\operatorname{d}(A, M) = \operatorname{d}(M, B), \quad \angle AOM = \angle MOB = \dato{m}{\frac{\theta}{2}}

    The triangle is isosceles; the perpendicular drawn from OO to the chord cuts it at its midpoint MM and divides the central angle into two equal angles, because the triangles with vertices OO, MM, AA and OO, MM, BB are right-angled at MM, have equal angles at AA and at BB, the triangle being isosceles, and share the side OMOM: by the two-angle similarity criterion they are similar with ratio 11.

  3. d⁡(M,B)=sin⁡θ2→C(θ)=2sin⁡θ2\operatorname{d}(M, B) = \resaltar{\sin \frac{\theta}{2}} \rightarrow \dato{c}{C(\theta) = \resaltar{2 \sin \frac{\theta}{2}}}

    In the right triangle with vertices OO, MM and BB, the hypotenuse measures 11 and the leg opposite the angle θ2\frac{\theta}{2} is half the chord; by the definition of the sine, that leg is sin⁡θ2\sin \frac{\theta}{2}.

  4. sin⁡α=12C(2α),60C(θ)=120sin⁡θ2\sin \alpha = \frac{1}{2} C(2\alpha), \qquad 60 C(\theta) = 120 \sin \frac{\theta}{2}

    Read backward, with θ=2α\theta = 2\alpha, the equality says that the sine of an angle is half the chord of the double angle; in the circle of radius 6060 used by Ptolemy all lengths are multiplied by 6060.

From the half arc to the sine

The astronomers of India, such as Aryabhata around the year 500, tabulated directly the half of the chord of the double arc, which they called jya, and thereby placed at the center of computation a leg of a right triangle instead of a chord. The word passed into Arabic as jiba, which in writing without vowels was confused with jaib, “fold” or “bosom” of a garment, and the Latin translators of the twelfth century rendered it as sinus, from which the current name derives. The astronomers of the Islamic world, between the ninth and the eleventh centuries, among them al-Battani and Abu al-Wafa, added the tables of the tangent and of the cotangent, which came from the study of the shadows of sundials; in the thirteenth century, Nasir al-Din al-Tusi set out trigonometry as a discipline independent of astronomy.

Trigonometry as a discipline

In Europe, Regiomontanus wrote around 1464 a treatise on triangles of every kind, printed in 1533, which brought together the solving of plane and spherical triangles; the word trigonometry, “measurement of triangles”, appears in the title of a work by Bartholomaeus Pitiscus from 1595. The law of cosines has a longer history: Euclid, in Book II of the Elements, had already corrected the Pythagorean theorem for acute and obtuse triangles by means of twice a rectangle, formed by one side and the projection of the other onto it, which today is written 2bccos⁡A2bc \cos A up to sign, and in the fifteenth century al-Kashi gave it a form suited to calculation. As for Heron’s formula, it appears in the Metrica of Heron of Alexandria, in the first century.

From ratios to functions

For centuries, the sine of an angle was a length in a circle whose radius was fixed by convention, at sixty or at several million, depending on the table. The infinitesimal calculus of the seventeenth century showed that the formulas become simpler if the radius is 11 and angles are measured by the length of their arc, and Leonhard Euler, in his Introductio in analysin infinitorum of 1748, treated the sine and the cosine as functions of a real variable, defined on the unit circle, which is the point of view of this unit; the name radian for the corresponding unit of measure was coined in the second half of the nineteenth century. At the beginning of the nineteenth century, Joseph Fourier maintained, in his analytical theory of heat published in 1822, that a very general periodic function can be written as a sum of sines and cosines, and the trigonometric functions became the language of all periodic phenomena.

Polar coordinates

The idea of describing a point by its distance to a center and by an angle is as old as astronomy, and Archimedes, in the third century before the Common Era, defined his spiral precisely in this way: a point that moves away from a center at constant speed while the ray carrying it rotates at constant speed. As a coordinate system, however, it appears in the seventeenth century: Bonaventura Cavalieri and Grégoire de Saint-Vincent used polar coordinates to study spirals, Isaac Newton included them among the systems with which he described curves, and Jacob Bernoulli employed them in his works from the end of that century; Euler systematized them in the eighteenth century. In the meantime, Johannes Kepler had published in 1609 that the orbit of Mars is an ellipse with the Sun at one of its foci, and the polar equation of the conics with a focus at the pole, which closes this unit, is the natural form of that law.

The problem that gave rise to them

The problem common to all these developments can be stated thus: given some angles and some lengths of a figure, calculate the others. Its simplest form is the right triangle. By similarity, the quotient of two sides depends only on the acute angle α\alpha, and is therefore a function of α\alpha; but that function is not algebraic, it is not obtained from α\alpha by sums, products or roots, and that is why it had to be tabulated. The example below, the first known attempt to measure the solar system, shows the problem in its pure state.

ExampleThe triangle of Aristarchus

When the Moon is seen exactly at quarter, the illuminated half of its disk is seen edge-on, and the Sun, the Moon and the Earth form a triangle with its right angle at the Moon. Aristarchus measured, from the Earth, the angle α\alpha between the directions of the Moon and of the Sun, and obtained 87∘87^\circ. Let TT, LL and SS be the Earth, the Moon and the Sun.

Proof

  1. d⁡(T,S)2=d⁡(T,L)2+d⁡(L,S)2{\operatorname{d}(T, S)}^2 = {\operatorname{d}(T, L)}^2 + {\operatorname{d}(L, S)}^2

    The triangle with vertices TT, LL and SS is right-angled at LL: the distance from the Earth to the Sun is the hypotenuse.

  2. d⁡(T,S)d⁡(T,L)=1cos⁡α=sec⁡α\frac{\operatorname{d}(T, S)}{\operatorname{d}(T, L)} = \resaltar{\frac{1}{\cos \alpha}} = \dato{s}{\resaltar{\sec \alpha}}

    The leg adjacent to the angle α\alpha is the distance to the Moon; by the definition of the cosine, the quotient of the distance to the Sun by the distance to the Moon depends only on α\alpha.

  3. α=87∘→18<sec⁡87∘<20\alpha = 87^\circ \rightarrow 18 < \sec 87^\circ < 20

    Without tables of cosines, Aristarchus bounded the quotient with geometric arguments and concluded that the Sun is between 1818 and 2020 times farther away than the Moon. The secant of 87∘87^\circ is approximately 19.1, within his bounds.

  4. α→90∘→sec⁡α→∞\alpha \to 90^\circ \rightarrow \sec \alpha \to \infty

    The method was correct, but the datum was not: the true angle differs from a right angle by less than a sixth of a degree, and the true quotient is around 390390. When α\alpha approaches a right angle, the Sun recedes without limit along the line that passes through the Moon, and a minute error in the angle produces an enormous error in the distance.

Since the Sun and the Moon have almost the same apparent size, as total solar eclipses show, the reasoning of Aristarchus also gave the ratio between their diameters, and with it the conclusion that the Sun is much larger than the Earth. The example contains in germ the whole unit: a ratio between sides that depends on an angle, the need to know that dependence precisely, and the observation that certain functions grow without bound near isolated values of the angle, which is the vertical asymptote of the secant studied in the lesson on the trigonometric functions and their graphs.

The same problem presents itself in two other ways, which give rise to the remaining lessons of the unit. The first is inverse: the ratio is known and the angle is sought, as when one measures the slope of a road and wants to know its inclination; answering it requires inverting functions that are not injective, and that is the task of the inverse trigonometric functions. The second is that of describing what rotates: a planet around the Sun, a wheel, a wave that repeats. For this it is convenient to name points by their distance to a center and their direction, and not by their distances to two axes, and to express periodic motion with functions that repeat; thus are born polar coordinates and sinusoids.

Relevance

In the rest of this course, unit 2, which begins with the lesson on the fundamental trigonometric identities, proves the formulas for the sine and the cosine of a sum, solves trigonometric equations in general and, by means of the rotation of axes, classifies all second-degree equations in two variables. Unit 3, devoted to complex numbers in polar form, writes each complex number with its modulus and its angle, as in the lesson on the polar form of complex numbers, and reduces powers and roots to multiplying moduli and adding angles; unit 4 represents the rotations of the plane by matrices whose entries are sines and cosines, in the lesson on matrices as transformations of the plane; and unit 5, on analytic geometry of space, measures the angle between two vectors in space by its cosine, in the lesson on vectors in space and the dot product, and uses it to calculate angles between lines and planes.

In other courses on the site, the inequality sin⁡x<x<tan⁡x\sin x < x < \tan x is the starting point for calculating the derivative of the sine in the course Calculus I, and arc lengths, areas bounded by polar curves and trigonometric substitutions are studied in the course Calculus II. The course Foundations of Mechanics resolves forces and velocities into components by means of the sine and the cosine, and studies circular motion through the relation between angular and linear velocity. In the course Programming in C++ I, the mathematical libraries of the language offer, besides the sine, the cosine and the arctangent, a function of two arguments that returns the polar angle of a point between −π-\pi and π\pi, with the same correction according to the quadrant that is proved in this unit.

Outside mathematics, trigonometry is the tool of surveying, navigation and geodesy, which measure the terrain by triangulation; polar coordinates are those of radars and of the systems that locate an object by its distance and its bearing; and sinusoids are the language of acoustics, alternating current, telecommunications and signal and image processing, which decompose every signal into simple oscillations following the idea of Fourier.