Active Prelude to Calculus

The sine and cosine functions, originally defined in the context of a point traversing the unit circle, are also central in right triangle trigonometry. They enable us to find missing information in right triangles in a straightforward way when we know one of the non-right angles and one of the three sides of the triangle, or two of the sides where one is the hypotenuse. In addition, we defined the tangent function in terms of the sine and cosine functions, and the tangent function offers additional options for finding missing information in right triangles. We’ve also seen how the inverses of the restricted sine, cosine, and tangent functions enable us to find missing angles in a wide variety of settings involving right triangles.

One of the powerful aspects of trigonometry is that the subject offers us the opportunity to view the same idea from many different perspectives. As one example, we have observed that the functions \(f(t) = \cos(t)\) and \(g(t) = \sin(t+\frac{\pi}{2})\) are actually the same function; as another, for \(t\) values in the domain \((-\frac{\pi}{2}, \frac{\pi}{2})\text{,}\) we know that writing \(y = \tan(t)\) is the same as writing \(t = \arctan(y)\text{.}\) Which perspective we choose to take often depends on context and given information.

While almost every question involving trigonometry can be answered using the sine, cosine, and tangent functions, sometimes it is convenient to use three related functions that are connected to the other three possible arrangements of ratios of sides in right triangles.

Note particularly that like the tangent function, the secant, cosecant, and cotangent are also defined completely in terms of the sine and cosine functions. In the context of a right triangle with an angle \(\theta\text{,}\) we know how to think of \(\sin(\theta)\text{,}\) \(\cos(\theta)\text{,}\) and \(\tan(\theta)\) as ratios of sides of the triangle. We can now do likewise with the other trigonometric functions:

With these three additional trigonometric functions, we now have expressions that address all six possible combinations of two sides of a right triangle in a ratio.

Because the sine and cosine functions are used to define each of the other four trigonometric functions, it follows that we can translate information known about the other functions back to information about the sine and cosine functions. For example, if we know that in a certain triangle \(\csc(\alpha) = \frac{5}{3}\text{,}\) it follows that \(\sin(\alpha) = \frac{3}{5}\text{.}\) From there we can reason in the usual way to determine missing information in the given triangle.

It’s also often possible to view given information in the context of the unit circle. With the earlier given information that \(\csc(\alpha) = \frac{5}{3}\text{,}\) it’s natural to view \(\alpha\) as being the angle in a right triangle that lies opposite a leg of length \(3\) with the hypotenuse being \(5\text{,}\) since \(\csc(\alpha) = \frac{\text{hyp}}{\text{opp}}\text{.}\) The Pythagorean Theorem then tells us the leg adjacent to \(\alpha\) has length \(4\text{,}\) as seen in \(\triangle OPQ\) in Figure 4.5.3.

But we could also view \(\sin(\alpha) = \frac{3}{5}\) as \(\sin(\alpha) = \frac{\frac{3}{5}}{1}\text{,}\) and thus think of the right triangle has having hypotenuse \(1\) and vertical leg \(\frac{3}{5}\text{.}\) This triangle is similar to the originally considered \(3\)-\(4\)-\(5\) right triangle, but can be viewed as lying within the unit circle. The perspective of the unit circle is particularly valuable when ratios such as \(\frac{\sqrt{3}}{2}\text{,}\) \(\frac{\sqrt{2}}{2}\text{,}\) and \(\frac{1}{2}\) arise in right triangles.

Like the tangent function, the secant, cosecant, and cotangent functions are defined in terms of the sine and cosine functions, so we can determine the exact values of these functions at each of the special points on the unit circle. In addition, we can use our understanding of the unit circle and the properties of the sine and cosine functions to determine key properties of these other trigonometric functions. We begin by investigating the secant function.

Using the fact that \(\sec(t) = \frac{1}{\cos(t)}\text{,}\) we note that anywhere \(\cos(t) = 0\text{,}\) the value of \(\sec(t)\) is undefined. We denote such instances in the following table by “u”. At all other points, the value of the secant function is simply the reciprocal of the cosine function’s value. Since \(|\cos(t)| \le 1\) for all \(t\text{,}\) it follows that \(|\sec(t)| \ge 1\) for all \(t\) (for which the secant’s value is defined).

Table 4.5.4 and Table 4.5.5 help us identify trends in the secant function. The sign of \(\sec(t)\) matches the sign of \(\cos(t)\) and thus is positive in Quadrant I, negative in Quadrant II, negative in Quadrant III, and positive in Quadrant IV.

In addition, we observe that as \(t\)-values in the first quadrant get closer to \(\frac{\pi}{2}\text{,}\) \(\cos(t)\) gets closer to \(0\) (while being always positive). Since the numerator of the secant function is always \(1\text{,}\) having its denominator approach \(0\) (while the denominator remains positive) means that \(\sec(t)\) increases without bound as \(t\) approaches \(\frac{\pi}{2}\) from the left side. Once \(t\) is slightly greater than \(\frac{\pi}{2}\) in Quadrant II, the value of \(\cos(t)\) is negative (and close to zero). This makes the value of \(\sec(t)\) decrease without bound (negative and getting further away from \(0\)) for \(t\) approaching \(\frac{\pi}{2}\) from the right side. We therefore see that \(p(t) = \sec(t)\) has a vertical asymptote at \(t = \frac{\pi}{2}\text{;}\) the periodicity and sign behavior of \(\cos(t)\) mean this asymptotic behavior of the secant function will repeat.

Plotting the data in the table along with the expected asymptotes and connecting the points intuitively, we see the graph of the secant function in Figure 4.5.6.

We see from both the table and the graph that the secant function has period \(P = 2\pi\text{.}\) We summarize our recent work as follows.

Trigonometric identities are simply identities that involve trigonometric functions. While there are a large number of such identities one can study, we choose to focus on those that turn out to be most useful in the study of calculus. The most important trigonometric identity is the fundamental trigonometric identity, which is a trigonometric restatement of the Pythagorean Theorem.

Identities are important because they enable us to view the same idea from multiple perspectives. For example, the fundamental trigonometric identity allows us to think of \(\cos^2(\theta) + \sin^2(\theta)\) as simply \(1\text{,}\) or alternatively, to view \(\cos^2(\theta)\) as the same quantity as \(1 - \sin^2(\theta)\text{.}\)

In calculus, it is also beneficial to know a couple of other standard identities for sums of angles or double angles. We simply state these identities without justification. For more information about them, see Section 10.4 in College Trigonometry, by Stitz and Zeager

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More information on Stitz and Zeager’s free texts can be found at http://stitz-zeager.com/.

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