**Chapter 9**

April 25 2017

**Chapter 9 Trigonometric Functions**
**9.1 Review of Trigonometry**
- we continue the program started in Chapter 8 of extending the scope of our work to include broader and broader classes of functions
- this time the trigonometric functions
- in science, these functions are indispensable tools for the study of periodic phenomena of all kinds
- in mathematics, as we shall see in Chapter 10, almost all of the more advanced methods of integration lean heavily on the trigonometric functions and their properties
**Radian Measure**
- the most common unit for measuring angles is the degree (1 right angle = 90 degrees)
- however, the standard unit for angle measurement in calculus is the radian
- one radian is the angle which, placed at the centre of a circle, subtends an arc whose length equals the radius
- more generally, the number of radians in an arbitrary central angle is defined to be the ratio of the length of the subtended arc to the radius
- angle = s / r
- equivalently, a central angle of angle radians subtends an arc of length angle times the radius
- s = angle r
- since the circumference of the circle is c = 2 pi r
- a complete central angle of 360 degrees is equivalent to 2 pi r / r = 2 pi radian
- just as the calculus of logarithms is simplified by using the base e
- the calculus of trigonometric functions is simplified by using radian measure
- we will point out the specific reason for this in section 9.2
- throughout our work we will use radian measure routinely and mention degrees only in passing
- it is sometimes useful to know that the area A of the section whose central angle is angle is
- A = 1/2 r s = 1/2 r2 angle, since s = r angle
- this is easy to remember by thinking of the section as if it were a triangle with height r and base s
**The Trigonometric Functions**
- consider the unit circle in the xy plane
- let the radius OP revolve counterclockwise through angle radians
- so angle = pi produces half a revolution and angle = 2 pi produces a complete revolution
- in this way, each real number angle determines a unique position of OP and a unique point P = (x, y) with the property that x2 + y2 = 1
- the sine and cosine of angle are now defined by:
- sin angle = y and cos angle = x
- it is evident from the definition that -1 <= sin angle <= 1
- and similarly for cos angle
- and the algebraic signs of these quantities depend on which quadrant the point P happens to lie in
- for every angle, the numbers angle and angle + 2 pi clearly determine the same point P
- so the values of sin angle and cos angle repeat when angle increases by 2 pi
- we express this property of sin angle and cos angle by saying that these functions are periodic with period 2 pi
- the remaining four trigonometric functions, the tangent, cotangent, secant, and cosecant, are defined by:
- tan angle = y / x, cot angle = x / y, sec angle = 1 / x, csc angle = 1 / y
- the sine and cosine are the basic functions
- the others can be expressed in terms of these two
- the right triangle interpretations of the trigonometric functions have many uses in physics and geometry
- nonetheless, the purposes of calculus require that angle be an unrestricted real variable
- for this reason, the unit circle definitions are preferable
**Identities**
- several simple relations among our functions are direct consequences of the definitions
- tan angle = sin angle / cos angle
- cot angle = cos angle / sin angle
- sec angle = 1 / cos angle
- csc angle = 1 / sin angle
- tan angle = 1 / cot angle
- altogether, there are 21 fundamental identities that express the main properties of the trigonometric functions and constitute the core of the subject
- these identities fall into several natural groups and are therefore easier to remember than we might expect
- our next identities state the effect of replacing angle with -angle
- sin (- angle) = - sin angle
- cos (- angle) = cos angle
- tan (- angle) = - tan angle
- our next group consists of three equivalent versions of the equation x2 + y2 = 1
- before stating these, we must explain that the symbols sin 2 angle is a standard notation for (sin angle) 2
- if we write x2 + y2 = 1 in the form y2 + x2 = 1
- this yields the first of the identities
- the others two are derived by dividing by cos 2 angle and sin 2 angle
- sin 2 angle + cos 2 angle = 1
- tan 2 angle + 1 = sec 2 angle
- 1 + cot 2 angle = csc 2 angle
- the following are called the addition formulas
- sin (a + b) = sin a cos b + cos a sin b
- cos (a + b) = cos a cos b - sin a sin b
- tan (a + b) = tan a + tan b / 1 - tan a tan b
- the corresponding subtraction formulas are
- sin (a - b) = sin a cos b - cos a sin b
- cos (a - b) = cos a cos b + sin a sin b
- tan (a - b) = tan a - tan b / 1 + tan a tan b
- the double-angle formulas are
- sin 2 a = 2 sin a cos a
- cos 2 a = cos 2 a - sin 2 a
- the half-angle formulas are
- 2 cos 2 a = 1 + cos 2 a
- 2 sin 2 a = 1 - cos 2 a
**Values**
- if we keep firmly in mind the definitions of sin a, cos a, and tan a
- then there are several first quadrant values of a for which the exact values of these functions are easy to find
- sin pi / 6 = 1/2, cos = 1/2 sqrt(3), tan = 1/3 sqrt(3)
- similarly for pi/4, pi/3, 0, pi/2, 3pi/2 and 2pi
- we also emphasize the way the algebraic signs of our functions vary from one quadrant to another
**Graphs**
- the graph of sin a is easy to sketch by following the way y varies as a increases from 0 to 2 pi
- that is, as the radius swings around through one complete counterclockwise revolution
- it is clear that sin a starts at 0, increases to 1, decreases to 0, decreases further to -1, and increases to 0
- this gives one complete cycle of sin a
- the complete graph consists of infinitely many repetitions of this cycle, to the right and to the left
- the graph of cos a can be sketched in essentially the same way, but shifted by pi / 2
- the graph of tan a is quite different
- we point out first that tan a is period with period pi
- tan (a + pi) = sin(a + pi) / cos(a + pi) = - sin a / - cos a = tan a
- the fact that tan a -> inf as a -> pi/2 from the left is often loosely expressed by writing tan pi/2 = inf
**Law of Cosines**
- this is a useful tool in a variety of situations in mathematics and physics
- it expresses the third side of a triangle in terms of two given sides a and b and the included angle a
- c2 = a2 + b2 - 2 a b cos a
**9.2 The Derivatives of The Sine and Cosine**
- the calculus of the trigonometric functions begins with two of the most important functions in all of mathematics
- d/dx sin x = cos x
- d/dx cos x = - sin x
- we emphasize that the letter x used here is simply an ordinary real variable, as in any function y = f(x)
- and if it is thought of as an angle, then this angle is always understood in radian measure
- the addition formulas for the sine and cosine obviously play essential roles in these arguments
- this is their main use in mathematics
- we now generalize by means of the chain rule
- and obtain the extremely useful formulas
- d/dx sin u = cos u du/dx
- d/dx cos u = - sin u du/dx
- as usual, u is understood to be any differentiable function of x
- students must learn to use these formulas in combination with all previous rules of differentiation
- in this connection it is necessary to remember the standard notation for powers of trigonometric functions
- sin n x means (sin x) n
- there is one exception to this usage
- for (sin x) -1 is never written sin -1 x
- the latter expression is reserved exclusively for the inverse sine function
- we are now able to explain why radian measure is preferred to degree measure when working with the trigonometric functions in calculus
- d/dx sin x deg = pi / 180 cos x deg
- we use radian measure routinely in calculus for the sake of simplicity
- in order to avoid the repeated occurrence of the factor pi/180
**9.3 The Integrals of the Sine and Cosine. The Needle Problem**
- the differential versions of the derivative functions in the previous section are
- d(cos u) = - sin u du
- d(sin u ) = cos u du
- these immediately yield the integration formulas
- S sin u du = - cos u + c
- S cos u du = sin u + c
- solve these types of integrations by substitution for u
**9.4 The Derivatives of The Other Four Functions**
- we can now complete our list of formulas for differentiating the trigonometric functions
- d/dx tan u = sec 2 u du/dx
- d/dx cot u = - csc 2 u du/dx
- d/dx sec u = sec u tan u du/dx
- d/dx csc u = - csc u cot u du/dx
- these formulas are quite easy to remember if we notice that the derivative of each cofunction (cot, csc) can be obtained from the derivative of the corresponding function (tan, sec) by
- inserting a minus sign and replacing each function by its cofunction
- the differentiation formulas immediately produce the associated integration formulas
- S sec 2 u du = tan u + c
- S csc 2 u du = - cot u + c
- S sec u tan u du = sec u + c
- S csc u cot u du = - csc u + c
**9.5 The Inverse Trigonometric Functions**
- our attention in this section is focused on the two integration formulas
- S dx / sqrt (1 - x2) = sin -1 x
- S dx / 1 + x2 = tan -1 x
- the unfamiliar functions on the right will be fully explained below
- they are the inverse trigonometric functions
- and are created expressly to enable us to calculate the integrals on the left
- these functions have other uses, but this is their primary purpose, the main justification for their existence
- before we start at the beginning and give a careful and orderly description of these functions
- we pause briefly to understand the rough way how they originate
- if we set x = sin angle
- dx = cos angle d angle
- S dx / sqrt (1 - x2) = S cos angle d angle / cos angle = S d angle = angle
- the process of solving x = sin angle for angle in terms of x is symbolized by writing
- angle = sin -1 x
- this makes the point about the way the inverse trigonometric functions arise
- they are forced upon us by the need to calculate certain integrals
**The Inverse Sine**
- we know that sin pi / 6 = 1/2
- so if we are asked to find an angle (in radian measure) whose sine is 1/2
- we can answer at once that pi / 6 is such an angle
- we are also aware that there are many other angles with this property
- as we have just seen, it is necessary in calculus to have a symbol to denote an angle whose sine is a given number y
- there are two such symbols in everyday use
- sin -1 x and arcsin x
- both mean an angle whose sine is x
- it is essential to understand that in the symbol sin -1 x
- the -1 is not an exponent
- so sin -1 x never means 1 / sin x
- so the formulas
- x = sin y and y = sin -1 x
- mean exactly the same thing
- in order to sketch the graph of y = sin -1 x
- it suffices to sketch x = sin y and rotate the axes
- to ensure for each x there is only one valid y
- the only values of y = sin -1 x we consider are those that lie in the interval - pi / 2 <= y <= pi / 2
- and this restriction is henceforth part of the meaning of the symbol y = sin -1 x
**The Inverse Tangent**
- the function y = tan -1 x is defined in essentially the same way
- y = tan -1 x means x = tan y and - pi / 2 < y < pi / 2
- the symbol tan -1 x is read the inverse tangent of x
- and it means the angle (in the specified interval) whose tangent is x
- we now calculate the derivative dy/dx of the function y = sin -1 x by differentiating
- x = sin y
- implicitly with respect to x
- the result is 1 = cos y dy/dx
- so dy/dx = 1 / cos y = 1 / sqrt (1 - sin 2 y) = 1 / sqrt (1 - x2)
- we choose the positive square root here because
- y = sin -1 x is clearly an increasing function
- this result can be written in the form
- d/dx sin -1 x = 1 / sqrt (1 - x2)
- when -1 < x < 1
- in just the same way we find the derivative of y = tan -1 x
- by differentiating x = tan y implicitly with respect to x
- this gives
- 1 = sec 2 y dy/dx
- so dy/dx = 1 / sec 2 y = 1 / 1 + tan 2 y = 1 / 1 + x2
- so we have
- d/dx tan -1 x = 1 / 1 + x2
- for all x
- these formulas are facts that lead to the main tools of this section
- first, we have the chain rule extensions of these formulas, which greatly broaden their scope
- d/dx sin -1 u = 1 / sqrt (1 - u2) du/dx
- d/dx tan -1 u = 1 / 1 + u2 du/dx
- as usual, u is understood to be any differentiable function of x
- even more important for our future work are the integration formulas
- S du / sqrt (1 - u2) = sin -1 u + c
- S du / 1 + u2 = tan -1 u + c
- these formulas are indispensable tools for integral calculus
- and all by themselves amply justify the study of trigonometry
**9.6 Simple Harmonic Motion. The Pendulum**
- the study of vibrations in the broad sense (waves, currents) is one of the most fundamental themes of physical science
- and in any such study sines and cosines play a central role
- one of the simplest types of vibrations occurs when an object or point moves back and forth along a straight line (x-axis)
- in such a way that its acceleration is always proportional to its position and is directed in the opposite sense
- d2x/dt2 = -kx, k > 0
- motion of this kind is called simple harmonic motion
- to emphasize that the constant k is positive, it is customary to write k = a2 with a > 0
- then d2x/dt2 + a2x =0
- it is easy to see that any function of the form
- x = A sin (at + b)
- satisfies this condition
- it is equally true, though not so easy to see
- that every nontrivial solution of the differential equation can be written in this form
- since the function sin(at + b) oscillates between -1 and 1
- the function oscillates between -|A| and |A|
- the number |A| is called the amplitude of the motion
- also, since the sine is periodic with period 2 pi
- sin(at + b) is period with period 2 pi / a
- because this is the amount t must increase in order to increase at + b by 2 pi
- this number T = 2 pi / a is called the period of the motion
- and is the time required for one complete cycle
- if we measure t in seconds
- then the number f of cycles per second satisfies the equation fT = 1
- so f = 1 / T = a / 2 pi
- this number is called the frequency of the motion
- another equivalent form of hte general solution that is often useful is
- x = A cos (at + b)
- since sin(angle + pi / 2) = cos angle
- there are two main interpretations of simple harmonic motion
- one geometric and one physical
- the geometric meaning can be understood by considering a point P that moves with constant angular velocity around a circle of radius A
- if this constant angular velocity is defined by a, then
- d angle / dt = a, so angle = at + b
- where b is the value of angle when t = 0
- if Q is the projection of P on the x-axis
- then its x-coordinate is x = A cos angle = A cos (at + b)
- this shows that Q moves back and forth along the x-axis in simple harmonic motion
- as P moves steadily around the circle in uniform circular motion
- and any simple harmonic motion can be visualized in this way
- the physical meaning appears when we think of the equation as describing the motion of a body of mass m rather than merely a point
- Newton's second law of motion says that F = ma
- so the equation becomes
- 1/m F = -kx, or F = -kmx
- a force of this kind is called a restoring force
- because its magnitude is proportional to the displacement x
- and it always acts to pull the body back toward the equilibrium position x = 0
**9.7 The Hyperbolic Functions**
- there are certain simple combinations of exponential functions that occur occasionally in applications
- to which the name hyperbolic functions has been given
- the hyperbolic sine and cosine are defined by
- sinh x = 1/2 ( ex - e-x)
- cosh = x = 1/2 (ex + e-x)
- these functions satisfy many identities that are very similar to corresponding identities satisfied by the trigonometric functions
- for instance cosh 2 x - sinh 2 x = 1
- also, their differentiation and integration properties are quite similar to those of trigonometric functions
- d/dx sinh x = cosh x
- d/dx coshx = sinh x
- however, one of the most important properties of the trigonometric functions
- that of periodicity
- is not possessed by any of the hyperbolic functions
- we mention these functions because students should at least know that they exist
- and we mention them here because of their analogies with the trigonometric functions
- however, we shall make no use of them in any of our future work
- so we now drop the subject without any further ado