Chapter 4

Trigonometry

4.1 What Use is Trigonometry?

Trigonometry, as its Greek and Latin roots suggest, is primarily the study of triangles. As we will see, circles also play a very prominent role. Trigonometric functions are, in fact, sometimes designated as circular functions. Figure 4.1 suggests how a clever caveman might determine the height y of a cliff by measuring the distance x to its base and the angle image that he has to look upward. This could be done by making a scale drawing but, more elegantly, using trigonometry, the height y is equal to image.

image

Figure 4.1 Stone-age trigonometry.

4.2 Geometry of Triangles

In this section we review some useful elementary results for triangles, most of them dating back to Euclid’s Elements. Figure 4.2 shows the essential elements of a triangle: the three sides, a, b, and c, the angles opposite them, A, B, and C, and the altitude h, perpendicular to the base b. There are also analogous altitudes perpendicular to a and c, not shown.

image

Figure 4.2 Sides, angles, and altitude of a triangle.

A signature feature of Euclidean space is that the angles of a triangle add up to exactly image:

image (4.1)

(By contrast, in spherical geometry, the sum of the angles is greater than image, while in hyperbolic geometry, it is less than image.)

The sides of a triangle always fulfill the triangle inequalities:

image (4.2)

Triangles can be classified according to the number of equal sides and angles, as shown on the top row of Figure 4.3. An equilateral triangle has three equal sides and three equal angles: a = b = c and A = B = C. An isosceles triangle has two equal sides and two equal angles, for example: image and image. Finally, a scalene triangle has three unequal sides and angles: image and image. A right triangle has one angle equal to image. An acute triangle has all three of its angles less than image, while an obtuse triangle has one angle greater than image. The last three categories are illustrated in the bottom row of Figure 4.3.

image

Figure 4.3 Classification of triangles.

The area of a triangle is equal to image. In terms of the variables shown in Figure 4.4,

image (4.3)

Another expression, known as Heron’s (or Hero’s) formula, computes the area in terms of its sides a, b, and c:

image (4.4)

where s = (a + b + c)/2, the semiperimeter of the triangle.

The bisectors of the three angle of a triangle intersect at a point known as the incenter, shown as a red1 point in Figure 4.4. It is the center of the circle inscribed within the triangle, known as its incircle, also colored in red. Similarly, the perpendicular bisectors of the three sides meet at another point, called the circumcenter, which is the center of the circumcircle, which circumscribes the triangle. These latter features are shown in blue. Only for an equilateral triangle do the incenter and circumcenter coincide. The three altitudes of a triangle also meet at a point, which corresponds to the centroid (or center of gravity).

Problem 4.2.1

The area of a trapezoid equals the product of its average width times its altitude:

image

Prove this by finding the sum of the areas of the red and blue triangles.

image

image

Figure 4.4 Incircle and circumcircle of triangle.

4.3 The Pythagorean Theorem

For a right triangle with sides a and b and hypotenuse c

image (4.5)

A pictorial proof of the theorem was given in Section 1.3. Albert Einstein, as a schoolboy, supposedly worked out his own proof of Pythagoras’ theorem. His line of reasoning follows, although we may have changed the names of the variables he used. Figure 4.5 shows a right triangle cut by a perpendicular dropped to the hypotenuse from the opposite vertex. This produces three similar triangles since they all have the same angles image, and 90°. Each length a, b, and c represents the hypotenuse of one of the triangles. Since the area of each similar triangle is proportional to the square of its corresponding hypotenuse, we can write

image (4.6)

The variables E might stand for area or extent (Erstreckung in German), while m is a proportionality constant (maybe mengenproportional). Since image, the ms cancel out and the result is Pythagoras’ theorem (4.5).

image

Figure 4.5 Einstein’s proof of Pythagoras’ theorem. The proof does assume, perhaps prematurely, that the angles of a triangle add up to 180°.

The preceding story is perhaps an alternative interpretation of the famous Einstein cartoon reproduced in Figure 4.6.

image

Figure 4.6 Einstein’s proof of Pythagoras’ theorem (?) ©2005 by Sidney Harris.

Surprisingly, an alternative proof of the Pythagorean theorem was published by James A. Garfield, the 20th president of the United States. He is speculated to have been the most mathematically knowledgable US president. As shown in Figure 4.7, three right triangles are arranged to form a trapezoid (turned sideways from its usual orientation). The area of the trapezoid is equal to (a + b)(a + b)/2. The areas of the blue,1 red, and purple triangles are, respectively, ab/2, ba/2, and image. Equating the two expressions gives

image

which reduces to image, giving Pythagoras’ theorem for both the red and blue triangles.

image

Figure 4.7 Garfield’s proof of the Pythagorean theorem.

The converse of Pythagoras’ theorem is also valid. If three lengths a, b, and c satisfy Eq. (4.5), then they must form a right triangle with c as the hypotenuse. This provides a handy way for carpenters to construct a right angle: just mark off the lengths 3, 4, and 5 in., then the angle between the first two is equal to image. There are, in fact, an infinite number of integer triples which satisfy image, beginning with {3, 4, 5}, {5, 12, 13}, {8, 15, 17}.

Analogous relations do not exist for powers of integers greater than 2. What has long been called “Fermat’s last theorem” states that

image (4.7)

has no nonzero integer solutions x, y, and z when image. This is the best known instance of a Diophantine equation, which involves only integers. Fermat wrote in his notes around 1630, “I have discovered a truly remarkable proof which this margin is too small to contain.” This turned out to have been a mischievous tease that took over three centuries to unravel. Some of the world’s most famous mathematicians have since struggled with the problem. These efforts were not entirely wasted since they stimulated significant advances in several mathematical fields including analytic number theory and algebraic geometry. Fermat’s conjecture, as it should have been called, was finally proven in 1993–1995 by the British mathematician Andrew Wiles, working at Princeton University. What we should now call the Fermat-Wiles theorem, took some 200 journal pages to present. (This whole book, let alone the margin, is too small to contain the proof!)

4.4 image in the Sky

The Babylonians (ca. 2400 BC) observed that the annual track of the Sun across the sky took approximately 360 days. Consequently, they divided its near-circular path into image, as a measure of each day’s progression. That is why we still count one spin around a circle as image and a right angle as image. This way of measuring angles is not very fundamental from a mathematical point of view, however. Mathematicians prefer to measure distance around the circumference of a circle in units of the radius, defined as 1 radian (rad). The Greeks designated the ratio of the circumference to the diameter, which is twice the radius, as image. Thus image corresponds to image radians. A semicircle, image, is image radians, while a right angle, image, is image radians. One radian equals approximately image. You should be very careful to set your scientific calculators to “radians” when doing most trigonometric manipulations.

It has been known since antiquity that image is approximately equal to 3. The Old Testament (II Chronicles 4:2) contains a passage describing the building of Solomon’s temple: “Also he made a molten sea of ten cubits from brim to brim, round in compass … and a line of thirty cubits did compass it round about.” This appears to imply that the ancient Hebrews used a value of image. Two regular hexagons inscribed in and circumscribed around a circle (Figure 4.8) establish that the value of image lies in the range image.

image

Figure 4.8 The perimeter of the inscribed hexagon equals three times the diameter of the circle. Noting that the two pink triangles are similar, you can show that the perimeter of the circumscribed hexagon equals image times the diameter. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

As a practical matter, the value of image could be determined with a tape measure wound around a circular object. In carpentry and sewing, an adequate approximation is image. (In 1897, the Indiana House of Representatives decreed by a vote of 67–0 that image should be simplified to exactly 3.2. The measure however never reached the floor of the Indiana Senate.) Accurate computed values of image are usually obtained from power series for inverse trigonometric functions. Since we haven’t introduced these yet, let us demonstrate a method for computing image which uses only the Pythagorean theorem. It was originally the idea of Archimedes to construct n-sided regular polygons inscribed in a unit circle (radius = 1). Then, as n becomes larger and larger, the perimeter of the polygon approaches the circumference of the circle. Denote the side of one such polygon by image. Then the perimeter equals image and an estimate for image, which we can call image, is equal to image. For example, the hexagon in Figure 4.8 gives image.

Figure 4.9 shows how to calculate the length of a side image of a regular polygon with 2n sides from image, that of a polygon with n sides. Both polygons are inscribed in the unit circle with OA = OB = OC = 1. image is a side of the n-gon, while image are sides of the 2n-gon. The segment image since the radius OC drawn to the new vertex C is a perpendicular bisector of side AB. Using the Pythagorean theorem, we can find two alternative expressions for the segment CD:

image

and

image

Equating these, we obtain

image (4.8)

which relates the values of image and image:

image (4.9)

Such an iterative evaluation was first carried out by Viète in 1593, starting with an inscribed square with image. This produced, for the first time, an actual formula for image, expressed as an infinite sequence of nested square roots:

image (4.10)

where the underbrace N indicates the total number of square-root signs. (For compactness, n − 1 has been replaced by N.) For a image-sided polygon (corresponding to N = 9) this procedure gives a value accurate to six significant figures:

image (4.11)

A rational approximation giving the same accuracy is image.

image

Figure 4.9 Detail of construction of an inscribed regular polygon with double the number of sides. OA = OB = OC = 1, the radius of a unit circle. AB represents a side image, while AC and CB are sides image.

It has long been a popular macho sport to calculate image to more and more decimal places, using methods which converge much faster than the formulas we consider. The current (2012) record is held by a Japanese supercomputer, which gives image to over 10 trillion digits. While such exercises might have little or no practical value, they serve as tests for increasingly powerful supercomputers. Sequences of digits from image can be used to generate random numbers (technically pseudorandom) for use in Monte Carlo computations and other simulations requiring random input.

Those of you taking certain physics or chemistry courses might appreciate a cute mnemonic giving image to 15 digits (3.141 592 653 589 79). Adapted from James Jeans, it runs “Now I need a drink, alcoholic of course, after the heavy sessions involving quantum mechanics.”

4.5 Sine and Cosine

Let us focus on one angle of a right triangle, designated by image in Figure 4.10. We designate the two perpendicular sides as being opposite and adjacent to the angle image. The sine and cosine are then defined as the ratios

image (4.12)

Later we will also deal with the tangent:

image (4.13)

A popular mnemonic for remembering which ratios go with which trigonometric functions is “SOHCAHTOA,” which might be the name of your make-believe Native American guide through the trigonometric forest.

image

Figure 4.10 Right triangle used to define trigonometric functions.

It is extremely instructive to represent the sine and cosine on the unit circle, shown in Figure 4.11, in which the hypotenuse corresponds to a radius equal to 1 unit. The circle is conveniently divided into four quadrants, I–IV, each with image varying over an interval of image radians from the range 0 to image. The lengths representing image are vertical lines on the unit circle while those representing image are horizontal. The cosine is closer to the angle—you might remember this by associating cosine with cozy up and sine with stand off. The functions shown in green have positive values, while those shown in red have negative values. Thus image is positive in quadrants I and II, negative in quadrants III and IV, while image is positive in quadrants I and IV, negative in quadrants II and III. It should also be clear from the diagram that, for real values of image, image and image can have values only in the range [−1, 1].

image

Figure 4.11 Unit circle showing image and image in each quadrant. Positive values of the functions are shown in green, negative, in red. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Note that any angle image less than 0 or greater than image is indistinguishable on the unit circle from one in the range 0 to image. The trigonometric functions are periodic in image and have the same values for any image with integer n. The functions image and image are plotted in Figure 4.12. These are commonly designated as sinusoidal functions or “sine waves.” Note that sine and cosine have the same shape, being just displaced from one another by image. It can be seen that

image (4.14)

The angle image is known as the complement of image. Equation (4.14) is equivalent to a catchy-sounding rule: The function of an angle is equal to the corresponding cofunction of its complement. Note that cosine and sine are even and odd functions, respectively:

image (4.15)

Whenever one of these functions goes through zero, the other has a local maximum at image or minimum at image. This follows easily from differential calculus, as we will show later. Pythagoras’ theorem translates to the fundamental trigonometric identity:

image (4.16)

Note that image and image are conventionally written image and image. They are not to be confused, of course, with image and image.

image

Figure 4.12 Plots of sine and cosine.

To compound the notational confusion, image and image are used to designate inverse trigonometric functions, also written image and image, respectively. These inverse functions are related by the following correspondences:

image (4.17)

image (4.18)

Since image and image are periodic functions, their inverse functions must be multivalued. For example, if image, it must likewise equal image, for image The principal value of image, sometimes designated image, is limited to the range image, corresponding to image. Analogously, the principal value of image, likewise designated image, lies in the range image. Graphs of image and image can be obtained by turning Figure 4.12 counterclockwise by image and then reflecting in the image-axis.

Several values of sine and cosine occur so frequently that they are worth remembering. Almost too obvious to mention,

image (4.19)

When image or image, so that (4.16) implies

image (4.20)

which is a factor well known in electrical engineering, in connection with the rms voltage and current of an AC circuit. An equilateral triangle with side 1, cut in half, gives a “30–60–90 triangle.” Since image and image, you can show using Pythagoras’ theorem that

image (4.21)

As image, in the first quadrant of Figure 4.11, the length of the line representing image approaches the magnitude of the arc of angle image—remember this is measured in radians. This implies a very useful approximation

image (4.22)

For a triangle of arbitrary shape, not limited to a right triangle, two important relations connecting the lengths a,b,c of the three sides with the magnitudes of their opposite angles image can be derived. In Figure 4.13, a perpendicular is drawn from any side to its opposite angle, say from angle A to side a. Call this length d. It can be seen that image and also that image. Analogous relations can be found involving a and A. The result is the law of sines:

image (4.23)

The little triangle to the right of line d has a base given by image. Therefore the base of the little triangle to the left of d equals image. Using image again, and applying Pythagoras’ theorem to the left-hand right triangle, we find

image (4.24)

The identity (4.16) simplifies the equation, leading to the law of cosines:

image (4.25)

By et cyc we mean that the result holds for all cyclic permutations image. The law of cosines is clearly a generalization of Pythagoras’ theorem, valid for all triangles. It reduces to Pythagoras’ theorem when image, so that image.

image

Figure 4.13 Diagram for law of sines and law of cosines.

4.6 Tangent and Secant

Additional subsidiary trigonometric functions can be defined in terms of sine and cosine. Consider just the first quadrant of the unit circle, redrawn in Figure 4.14. Let the horizontal line containing image, as well as the hypotenuse, be extended until they intersect the circle. A vertical tangent line of length image can then be used to construct a larger right triangle. The new hypotenuse is called a secant and labeled image. Clearly the new triangle is similar to the original one, with its horizontal side being equal to 1, the unit-circle radius. Therefore

image (4.26)

Also by Pythagoras’ theorem,

image (4.27)

so that

image (4.28)

making use of Eq. (4.16). The tangent and secant are plotted in Figure 4.14. Since image, we find image, since the secant is the reciprocal of the cosine. Note that image is periodic in image, rather than image, and can take any value from image to image.

image

Figure 4.14 Graphs of tangent (red) and secant (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

It is also possible to define cofunctions in analogy with (4.14), namely the cotangent and cosecant:

image (4.29)

This completes the list of standard trigonometric functions. In terms of sine and cosine:

image (4.30)

4.7 Trigonometry in the Complex Plane

In Figure 3.4, a complex quantity z was represented by its Cartesian coordinates x and y. Alternatively, a point in the x-, y-plane can be represented in polar coordinates, usually designated r and image. Conventionally, r is the distance from the origin, while image is the angle that the vector image makes with the positive x-axis. The coordinates in the two systems are related by

image (4.31)

A complex quantity image expressed in polar coordinates is called a phasor. In place of r we have the modulus image. The angle image, now called the phase or the argument, plays the same role as in Figure 4.11, with the unit circle generalized to a circle of radius image. Thus the phasor representation of a complex number takes the form

image (4.32)

A convenient abbreviation which we will employ for a short while is

image (4.33)

In Figure 3.5 it was shown how multiplication of a complex numbers by i could be represented by a 90° rotation in the complex plane. Note that i can be represented by the phasor

image (4.34)

while

image (4.35)

We can generalize that the product of two arbitrary phasors, say image and image, is given by

image (4.36)

obtained by multiplying the two moduli while adding the two phases, as shown in Figure 4.15.

image

Figure 4.15 Multiplication of phasors.

Consider the multiplication of the two phasors image and image to give image. Written out in full, this gives

image (4.37)

Equating the separate real and imaginary parts of each side of the equation, we obtain the two fundamental angle-sum trigonometric identities:

image (4.38)

and

image (4.39)

From (4.38) and (4.39) we can derive the corresponding relation for image:

image (4.40)

Dividing both numerator and denominator by image and introducing image and image, we obtain

image (4.41)

Problem 4.7.1

As a special case of Eqs. (4.38) and (4.39), derive formulas for image and image:

image (4.42)

image (4.43)

Problem 4.7.2

Using the substitution image, derive the half-angle formulas:

image (4.44)

image (4.45)

4.8 de Moivre’s Theorem

Equation (4.36) can be applied to the square of a phasor image, of modulus 1, giving image. This can, in fact, be extended to the nth power of z giving image. This is a famous result known as de Moivre’s theorem, which we write out in full:

image (4.46)

Beginning with de Moivre’s theorem, useful identities involving sines and cosines can be derived. For example, setting image,

image (4.47)

Equating the real and imaginary parts on each side of the equation, we obtain the two identities

image (4.48)

and

image (4.49)

Analogously, with image in (4.46) we can derive:

image (4.50)

Problem 4.8.1

Verify the expressions for image and image.

de Moivre’s theorem can be used to determine the nth roots of unity, namely the n complex roots of the equation

image (4.51)

Setting image with image in Eq. (4.46), we find

image (4.52)

But, for integer k, image while image. Thus the nth roots of unity are given by

image (4.53)

When the nth roots of unity are plotted on the complex plane, they form a regular polygon with n sides, with one vertex at 1. For example, for image, for image, for image. The sum of the roots for each n adds to zero.

4.9 Euler’s Theorem

de Moivre’s theorem, Eq. (4.46), remains valid even for noninteger values of n. Replacing n by 1/m we can write

image (4.54)

or

image (4.55)

In the limit as image. At the same time image, as noted in Eq. (4.22). We can therefore write

image (4.56)

The limit defines the exponential function, as shown in Eq. (3.103). We arrive thereby at a truly amazing relationship:

image (4.57)

known as Euler’s theorem. (This is actually one of at least 13 theorems, formulas, and equations which goes by this name. Euler was very prolific!)

A very notable special case of Eq. (4.57), for image, is

image (4.58)

an unexpectedly simple connection between the three mathematical entities, e, i, and image, each of which took us an entire section to introduce. This result can also be rearranged to

image (4.59)

sometimes called Euler’s identity. Several authors regard this as the most beautiful equation in all of mathematics. It contains what are perhaps the five most fundamental mathematical quantities: in addition to e, i, and image, the additive identity 0 and the multiplicative identity 1. It also makes use of the concepts of addition, multiplication, exponentiation, and equality. Because it represents so much in one small package, the formula has been imprinted on the side of some far-ranging NASA spacecraft to demonstrate the existence of intelligent life on Earth. It is to be hoped, of course, that extraterrestrials would be able to figure out that those symbols represent a mathematical equation and not a threat of interstellar war.

Solving (4.57) and its complex-conjugate equation image for image and image, we can represent these trigonometric functions in terms of complex exponentials:

image (4.60)

and

image (4.61)

The power-series expansion for the exponential function given in Eq. (3.104) is, as we have noted, valid even for imaginary values of the exponent. Replacing x by image we obtain

image (4.62)

Successive powers of i are given sequentially by

image (4.63)

Collecting the real and imaginary parts of (4.62) and comparing with the corresponding terms in Euler’s theorem (4.57) result in power-series expansions for the sine and cosine:

image (4.64)

and

image (4.65)

4.10 Hyperbolic Functions

Hyperbolic functions are “copy-cats” of the corresponding trigonometric functions, in which the complex exponentials in Eqs. (4.60) and (4.61) are replaced by real exponential functions. The hyperbolic sine and hyperbolic cosine are defined, respectively, by

image (4.66)

Actually, hyperbolic functions result when sine and cosine are given imaginary arguments. Thus

image (4.67)

The hyperbolic sine and cosine functions are plotted in Figure 4.16. Unlike their trigonometric analogs, they are not periodic functions and both have the domains image. Note that as image both image and image approach image. The hyperbolic cosine represents the shape of a flexible wire or chain hanging from two fixed points, called a catenary (from the Latin catena = chain).

image

Figure 4.16 Hyperbolic sine and cosine.

By solving Eq. (4.66) for image and image, we obtain the analog of Euler’s theorem for hyperbolic functions:

image (4.68)

The identity image then leads to the hyperbolic analog of (4.16):

image (4.69)

The trigonometric sine and cosine are called circular functions because of their geometrical representation using the unit circle image. The hyperbolic functions can analogously be based on the geometry of the unit hyperbolaimage. We will develop the properties of hyperbolas, and other conic sections, in more detail in the following chapter. It will suffice for now to show the analogy with circular functions. Figure 4.17 shows the first quadrant of the unit circle and the unit hyperbola, each with specific areas image and image, respectively, shown shaded. For the circle, the area is equal to the fraction image of image, the area of the unit circle. Thus

image (4.70)

For the unit hyperbola, we will be able to compute the area image later using calculus. It will suffice for now to define a variable

image (4.71)

Analogous constructions in Figure 4.17 can then be used to represent the trigonometric functions image and the hyperbolic functions image.

image

Figure 4.17 Geometric representation of circular and hyperbolic functions. The argument of each function equals twice the corresponding shaded area.

The series expansions for the hyperbolic functions are similar to (4.64) and (4.65), except that all terms have plus signs:

image (4.72)

and

image (4.73)

Problem 4.10.1

Derive the analogs of Eqs. (4.38) and (4.39) for hyperbolic sine and cosine.


1For interpretation of the references to color in this Figures 4.4 and 4.7 legend, the reader is referred to the web version of this book.

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