LESSON 10: Graphs of the Trigonometric Functions

In this section, we return to our discussion of the circular (trigonometric) functions as functions of real numbers and pick up where we left off in Sections 10.2.1 and 10.3.1. As usual, we begin our study with the functions f(t) = cos(t) and g(t) = sin(t). 10.5.1 Graphs of the Cosine and Sine Functions From Theorem 10.5 in Section 10.2.1, we know that the domain of f(t) = cos(t) and of g(t) = sin(t) is all real numbers, (−∞, ∞), and the range of both functions is [−1, 1]. The Even / Odd Identities in Theorem 10.12 tell us cos(−t) = cos(t) for all real numbers t and sin(−t) = − sin(t) for all real numbers t. This means f(t) = cos(t) is an even function, while g(t) = sin(t) is an odd function.1 Another important property of these functions is that for coterminal angles α and β, cos(α) = cos(β) and sin(α) = sin(β). Said differently, cos(t+2πk) = cos(t) and sin(t+2πk) = sin(t) for all real numbers t and any integer k. This last property is given a special name. Definition 10.3. Periodic Functions: A function f is said to be periodic if there is a real number c so that f(t + c) = f(t) for all real numbers t in the domain of f. The smallest positive number p for which f(t+p) = f(t) for all real numbers t in the domain of f, if it exists, is called the period of f. We have already seen a family of periodic functions in Section 2.1: the constant functions. However, despite being periodic a constant function has no period. (We’ll leave that odd gem as an exercise for you.) Returning to the circular functions, we see that by Definition 10.3, f(t) = cos(t) is periodic, since cos(t + 2πk) = cos(t) for any integer k. To determine the period of f, we need to find the smallest real number p so that f(t + p) = f(t) for all real numbers t or, said differently, the smallest positive real number p such that cos(t + p) = cos(t) for all real numbers t. We know that cos(t + 2π) = cos(t) for all real numbers t but the question remains if any smaller real number will do the trick. Suppose p > 0 and cos(t + p) = cos(t) for all real numbers t. Then, in particular, cos(0 + p) = cos(0) so that cos(p) = 1. From this we know p is a multiple of 2π and, since the smallest positive multiple of 2π is 2π itself, we have the result. Similarly, we can show g(t) = sin(t) is also periodic with 2π as its period.2 Having period 2π essentially means that we can completely understand everything about the functions f(t) = cos(t) and g(t) = sin(t) by studying one interval of length 2π, say [0, 2π].3 One last property of the functions f(t) = cos(t) and g(t) = sin(t) is worth pointing out: both of these functions are continuous and smooth. Recall from Section 3.1 that geometrically this means the graphs of the cosine and sine functions have no jumps, gaps, holes in the graph, asymptotes, 1See section 1.6 for a review of these concepts. 2Alternatively, we can use the Cofunction Identities in Theorem 10.14 to show that g(t) = sin(t) is periodic with period 2π since g(t) = sin(t) = cos π 2 − t = f π 2 − t . 3Technically, we should study the interval [0, 2π),4 since whatever happens at t = 2π is the same as what happens at t = 0. As we will see shortly, t = 2π gives us an extra ‘check’ when we go to graph these functions. 4 In some advanced texts, the interval of choice is [−π, π]

corners or cusps. As we shall see, the graphs of both f(t) = cos(t) and g(t) = sin(t) meander nicely and don’t cause any trouble. We summarize these facts in the following theorem. Theorem 10.22. Properties of the Cosine and Sine Functions • The function f(x) = cos(x) • The function g(x) = sin(x) – has domain (−∞, ∞) – has domain (−∞, ∞) – has range [−1, 1] – has range [−1, 1] – is continuous and smooth – is continuous and smooth – is even – is odd – has period 2π – has period 2π In the chart above, we followed the convention established in Section 1.6 and used x as the independent variable and y as the dependent variable.5 This allows us to turn our attention to graphing the cosine and sine functions in the Cartesian Plane. To graph y = cos(x), we make a table as we did in Section 1.6 using some of the ‘common values’ of x in the interval [0, 2π]. This generates a portion of the cosine graph, which we call the ‘fundamental cycle’ of y = cos(x).

the movement of some key points on the original graphs. We choose to track the values x = 0, π 2 , π, 3π 2 and 2π. These ‘quarter marks’ correspond to quadrantal angles, and as such, mark the location of the zeros and the local extrema of these functions over exactly one period. Before we begin our next example, we need to review the concept of the ‘argument’ of a function as first introduced in Section 1.4. For the function f(x) = 1 − 5 cos(2x − π), the argument of f is x. We shall have occasion, however, to refer to the argument of the cosine, which in this case is 2x − π. Loosely stated, the argument of a trigonometric function is the expression ‘inside’ the function. Example 10.5.1. Graph one cycle of the following functions. State the period of each. 1. f(x) = 3 cos πx−π 2 + 1 2. g(x) = 1 2 sin(π − 2x) + 3 2 Solution. 1. We set the argument of the cosine, πx−π 2 , equal to each of the values: 0, π 2 , π, 3π 2 , 2π and solve for x.

 Next, we substitute each of these x values into f(x) = 3 cos πx−π 2 + 1 to determine the corresponding y-values and connect the dots in a pleasing wavelike fashion

 2. Proceeding as above, we set the argument of the sine, π − 2x, equal to each of our quarter marks and solve for x.

= π. The functions in Example 10.5.1 are examples of sinusoids. Roughly speaking, a sinusoid is the result of taking the basic graph of f(x) = cos(x) or g(x) = sin(x) and performing any of the transformations6 mentioned in Section 1.7. Sinusoids can be characterized by four properties: period, amplitude, phase shift and vertical shift. We have already discussed period, that is, how long it takes for the sinusoid to complete one cycle. The standard period of both f(x) = cos(x) and g(x) = sin(x) is 2π, but horizontal scalings will change the period of the resulting sinusoid. The amplitude of the sinusoid is a measure of how ‘tall’ the wave is, as indicated in the figure below. The amplitude of the standard cosine and sine functions is 1, but vertical scalings can alter this. 6We have already seen how the Even/Odd and Cofunction Identities can be used to rewrite g(x) = sin(x) as a transformed version of f(x) = cos(x), so of course, the reverse is true: f(x) = cos(x) can be written as a transformed version of g(x) = sin(x). The authors have seen some instances where sinusoids are always converted to cosine functions while in other disciplines, the sinusoids are always written in terms of sine functions. We will discuss the applications of sinusoids in greater detail in Chapter 11. Until then, we will keep our options open.

 The phase shift of the sinusoid is the horizontal shift experienced by the fundamental cycle. We have seen that a phase (horizontal) shift of π 2 to the right takes f(x) = cos(x) to g(x) = sin(x) since cos x − π 2 = sin(x). As the reader can verify, a phase shift of π 2 to the left takes g(x) = sin(x) to f(x) = cos(x). The vertical shift of a sinusoid is exactly the same as the vertical shifts in Section 1.7. In most contexts, the vertical shift of a sinusoid is assumed to be 0, but we state the more general case below. The following theorem, which is reminiscent of Theorem 1.7 in Section 1.7, shows how to find these four fundamental quantities from the formula of the given sinusoid. Theorem 10.23. For ω > 0, the functions C(x) = A cos(ωx + φ) + B and S(x) = A sin(ωx + φ) + B ˆ have period 2π ω ˆ have amplitude |A| ˆ have phase shift − φ ω ˆ have vertical shift B We note that in some scientific and engineering circles, the quantity φ mentioned in Theorem 10.23 is called the phase of the sinusoid. Since our interest in this book is primarily with graphing sinusoids, we focus our attention on the horizontal shift − φ ω induced by φ. The proof of Theorem 10.23 is a direct application of Theorem 1.7 in Section 1.7 and is left to the reader. The parameter ω, which is stipulated to be positive, is called the (angular) frequency of the sinusoid and is the number of cycles the sinusoid completes over a 2π interval. We can always ensure ω > 0 using the Even/Odd Identities.7 We now test out Theorem 10.23 using the functions f and g featured in Example 10.5.1. First, we write f(x) in the form prescribed in Theorem 10.23, f(x) = 3 cos πx − π 2 + 1 = 3 cos π 2 x + − π 2 + 1, 7Try using the formulas in Theorem 10.23 applied to C(x) = cos(−x + π) to see why we need ω > 0.