9.2
Basic Formulas for the Trigonometric
Functions.
Introduction.
The aim of this section is to list the basic formulas for the various trigonometric functions and learn their use. The actual theoretical development of these will be done later.
These functions, though easy to define, don’t have simple formulas and evaluating their values requires careful approximations. You can use a calculator and get decimal numbers, but that is not our goal.
We shall try to understand how these functions can be evaluated precisely for many special values of the argument and how they can be manipulated formally using algebraic techniques. They form the basis of many mathematical theories and are, in some sense, the most important functions of modern mathematics, next to
polynomials. Indeed, the theory of Fourier analysis lets us approximate all practical functions in terms of suitable trigonometric functions and it is crucial to learn how to manipulate them.
It is customary to also learn about exponential and logarithmic functions. Our discussion of Euler’s representation also indicates a formula for the exponential function
exp(x) = 1 + x
1!+ · · · + xn
n! + · · · . We shall discuss it later.
As far as evaluation at specific points is concerned, trigonometric, exponential or logarithmic functions are equally easy - they are keys on the calculator! You probably have already seen and used them in concrete problems anyway.
We have already described connection between the angle t, its corresponding locator point P (t) and the corresponding trigonometric functions
sin(t) = the y-coordinate of P (t) and cos(t) = the x-coordinate of P (t). Unless otherwise stated, our angles will always be in radian measure, so a right angle has measure π/2.
Negative angles.
Sometimes it reduces our work if we use negative values of angles, rather that converting them to positive angles. After all, −30◦ is easier to visualize than 330◦.
We can give a natural meaning to negative angles, they represent arc length
measure in the negative or clockwise direction! Check out that going 30◦ clockwise
or 330◦ counterclockwise leads to the same locator point!
Thus, sometimes it is considered more convenient to take the angle measure between −180◦ to 180◦.
158 CHAPTER 9. TRIGONOMETRY
that its locator point P (t) has coordinates (cos(t), sin(t)).
We now define four more functions which are simply related to the above two functions: tan(t) = sin(t) cos(t) , cot(t) = cos(t) sin(t). Also: csc(t) = 1 sin(t) , sec(t) = 1 cos(t).
These functions have full names which are:
Notation sin cos tan cot csc sec
Full name Sine Cosine Tangent Cotangent Cosecant Secant
We have the following basic identities derived from the fact that P (t) is on the unit circle.
Fundamental Identity 1. sin2(t) + cos2(t) = 1
We can divide this identity by sin2(t) or cos2(t) to derive two new identities for the
other functions. The reader should verify these:
Fundamental Identity 2. 1 + cot2(t) = csc2(t)
Fundamental Identity 3. tan2(t) + 1 = sec2(t)
Remark on notation. It is customary to write sin2(t) in place of (sin(t))2. This is
not done for other functions. Indeed, many times the notation f2(x) is used to
indicate applying the function twice - i.e. f ◦ f(x) = f(f(x)). It is wise to verify what this notation means before making guesses!
For trigonometric functions, the convention is as we explained above! If you have any confusion in your mind, it is safer to use the explicit (sin(t))2 rather than sin2(t).
This convention works for other powers too. This indeed leads to a confusion later on if you try to use negative powers. Here is how it comes up.
What is that angle?
We have given the definition of the values of trigonometric functions, given an angle either in degrees or radians. In may cases, we have no recourse but to use a
calculator to get an approximate answer.
Given a value a = sin(t) we ask what is t? We already know that there is no unique answer, since adding or subtracting multiples of 2π to any one answer leads to the same value of sin(t).
If we think of sin as a function from the real numbers to real numbers, we can observe the following:
9.2. BASIC FORMULAS FOR THE TRIGONOMETRIC FUNCTIONS. 159
• The function is not onto. In fact, the range is easily seen to be the interval [−1, 1], in view of the definition in terms of the circle.
• The function is not one to one. If we think of tracing the circle from t = −π/2 to π/2 counterclockwise, then it is not hard to see that every value of a
between −1 and 1 is reached once and exactly once.
The idea is that if we draw a line y = a, then it hits the right half of the circle in exactly one point. We can take t to be the angle corresponding to that point, keeping it between −π/2 and π/2, then we get what we want. • Thus, if we trim the domain of the sin function to [−π/2, π/2], then it is a one
to one and onto function with range [−1, 1].
This sine function has an inverse function, which we denote by sin−1 or arcsin.
Thus we have defined:
t = sin−1(a) if sin(t) = a and − π/2 ≤ t ≤ π/2.
• Now comes the confusion. It is tempting to think that sin−1(a) as the same as (sin(a))−1 = 1
sin(a). This is natural since we write sin2(t) for (sin(t))2. It is customary to avoid this confusion by using a different notation, arcsin in place of sin−1.
We will systematically use this notation.
• Complete solution of sin(t) = a. If t = arcsin(a), then clearly t is a solution of the equation sin(t) = a. There is, however another equally interesting solution, namely π − t. We will soon show that sin(π − t) = sin(t) for all values of t -the Supplementary Angle Identity. It corresponds to the other point of intersection of the line y = a with the circle x2+ y2 = 1.
Thus, we have two solutions: t = sin−1(a), π − sin−1(a). It can be shown that
all possible solutions are obtained from adding arbitrary multiples of 2π to these two basic answers.
We similarly have inverse function definitions for other trigonometric functions. We list these without further comments.
• Inverse cosine.
We define cos−1(a) = arccos(a) as the angle t such that cos(t) = a and
160 CHAPTER 9. TRIGONOMETRY
In general, there are two solutions to cos(t) = a, namely t = arccos(a) and t = − arccos(a). This follows from the evenness of the cosine function that we shall soon prove.
These also represent the two points of intersection of the vertical line x = a with the circle x2+ y2 = 1.
• Inverse tangent.
We define tan−1(a) = arctan(a) as the angle t such that tan(t) = a and
−π/2 ≤ t ≤ π/2.
In general, there are two solutions to tan(t) = a, namely t = arctan(a) and t = π + arctan(a). This follows from the Fundamental Identity 10 (to be proved below) by taking s = π.
• The remaining three inverse functions are defined naturally from above, but generally are not specifically needed. Thus sin(t) = a is the same as
csc(t) = 1
a. So, arccsc(a) is simply defined as arcsin( 1 a).
The others are similarly handled.