| Trigonometry Demystified, 2nd edition |
| Stan Gibilisco |
| Explanations for Quiz Answers in Chapter 5 |
| 1. Let's look at the graph of the hyperbolic Arctangent function (Fig. 5-10). When we start at the point on the curve where x = 0 and move to the right (increasing x), the value of y, which equals Arctanh x, skyrockets. As x approaches 1, y increases without limit in the positive direction. The correct choice is C. |
| 2. When we examine the graph of the hyperbolic tangent function (Fig. 5-4), we see that the range corresponds to the open interval (-1,1). As x increases negatively without limit, the value of y, which equals tanh x, approaches -1. The correct choice is A. |
| 3. Remember the formula for the hyperbolic cosine of a quantity in terms of
exponential functions. That formula, once again, is cosh x = (ex + e-x) / 2 If x = 1 (exactly), then we have cosh 1 = (e1 + e-1) / 2 where 1, -1, and 2 are exact values. When we use a calculator to evaluate these exponential expressions to five decimal places (more than enough, given the choices in this question), we get e1 = 2.71828 and e-1 = 0.36788 Therefore cosh 1 = (2.71828 + 0.36788) / 2 which rounds off to cosh 1 = 1.543 accurate to three decimal places. The correct choice is A. |
| 4. The natural exponential function and the natural logarithm function are mutual inverses. When we realize this fact, noting also that e = e1 because any quantity raised to the first power equals that quantity itself, we can surmise that ln e = ln e1 = 1. (If you like, you can use a calculator to check.) The correct choice is C. |
| 5. By definition, the hyperbolic cotangent equals the hyperbolic cosine divided by the
hyperbolic sine, assuming that the hyperbolic sine doesn't equal 0. In mathematical terms,
we write that fact as coth x = cosh x / sinh x The correct choice is D. |
| 6. A so-called hyperbolic angle comprises the area of an enclosed region adjacent to a unit hyperbola in Cartesian coordinates, as shown in Fig. 5-1 and described in the accompanying text. The answer is C. |
| 7. We can calculate the hyperbolic Arcsine of any real number x using the
formula
Arcsinh x = ln [x + (x2 + 1)1/2] where ln represents the natural logarithm function. When we input the value -4.0000 for x to the above formula, we get Arcsinh (-4.0000) = ln {-4.0000 + [(-4.0000)2 + 1]1/2} The correct choice is B. |
| 8. Figure 5-5 is a graph of the hyperbolic cosecant function. It appears to show that
as x increases positively without limit, the value of csch x approaches 0. We
can verify this fact by plugging progressively larger values of x into the
exponential formula
csch x = 2 / (ex - e-x) As x grows positively larger without limit, the value of ex also grows positively larger without limit, while the value of e-x approaches 0. Therefore, the quantity (ex - e-x) grows positively larger without limit, so 2 divided by the quantity (ex - e-x) must approach 0. The correct choice is B. |
| 9. When we examine the graph of the hyperbolic sine function (Fig. 5-2), it appears
that the curve exhibits symmetry with respect to the origin, so it's tempting to suppose
that sinh -3 = -(sinh 3) and the two quantities thus add up to 0. To gain absolute confidence in this conclusion, let's use the exponential formula for the hyperbolic sine. We can derive sinh -3 = [e-3 - e-(-3)] / 2 The two values are negatives of each other, so they add up to 0. The correct choice is B. |
| 10. When we look at the graph of the hyperbolic cosine function (Fig. 5-3), it appears
that the curve is symmetrical relative to the y (vertical) axis, so it makes sense
to imagine that
cosh -3 = cosh 3 To confirm this notion, let's compare the values using the exponential formula for the hyperbolic cosine. We have cosh -3 = [e-3 + e-(-3)] / 2 The two values equal each other. The correct choice is D. |