Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a calculator to approximate each value. $$ \sec ^{-1}(-2.222) $$

Step-by-Step Solution

Answer

\( \sec^{-1}(-2.222) \approx 2.034 \, \text{radians} \).

Step 1: Understand the Inverse Secant Function

The inverse secant function, denoted as \( \sec^{-1}(x) \), is the inverse of the secant function. It returns the angle \( \theta \) such that \( \sec(\theta) = x \). The range of \( \sec^{-1}(x) \) is \([0, \pi]\) excluding \( \frac{\pi}{2} \).

Step 2: Set Up for Calculator Input

For using a calculator, note that many calculators provide the arcsine, arccosine, and arctangent directly. Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we can find the angle by computing \( \cos^{-1}\left(\frac{1}{x}\right) \). Here, compute \( \cos^{-1}\left(\frac{1}{-2.222}\right) \).

Step 3: Calculate \( \frac{1}{-2.222} \)

Calculate \( \frac{1}{-2.222} \) using a calculator. This gives approximately \(-0.450\).

Step 4: Compute \( \cos^{-1}(-0.450) \)

Use the calculator to find \( \cos^{-1}(-0.450) \). This gives the angle whose cosine is \(-0.450\). From the calculator, \( \cos^{-1}(-0.450) \) is approximately \( \, 2.034\) radians.

Key Concepts

Secant Function

The secant function, denoted as \( \sec(\theta) \), is a trigonometric function that is closely related to the cosine function. In fact, it is defined as the reciprocal of the cosine function. This means:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

This definition makes the secant function particularly useful for working with trigonometric equations in which dividing by cosine might simplify the expression. In the context of inverse trigonometric functions like \( \sec^{-1}(x) \), this understanding helps in calculating angles such that the secant of those angles is \( x \).
The inverse secant function, therefore, gives an angle as output when you input a ratio of the hypotenuse to the adjacent side of a right triangle or any real-valued number outside the interval [-1, 1]. This is because for values between -1 and 1, the reciprocal cosine would lead to undefined secant values. Remembering these aspects of the secant function is essential, especially when working with calculus problems or real-world applications that involve wave motion, optics, or electrical engineering.

Arccosine Calculation

The arccosine function, indicated by \( \cos^{-1}(x) \), is another inverse trigonometric function used to find an angle whose cosine is \( x \). Its principal range is \([0, \pi]\), which means any angle it returns will be within this interval. To find \( \sec^{-1}(-2.222) \), you use the relationship between secant and cosine:

• First, convert the secant function problem to a cosine problem: \( \cos^{-1}\left(\frac{1}{x}\right) \)
• In this exercise, replace \( x \) with -2.222. Calculate \( \frac{1}{-2.222} \) giving approximately -0.450.
• Now, compute the arccosine: \( \cos^{-1}(-0.450) \).

The process ensures that the exact angle can be found using a calculator, which provides the needed trigonometric values. In this scenario, \( \cos^{-1}(-0.450) \) approximately equals 2.034 radians. Such calculations are crucial when analyzing physics problems involving circular motion or engineering challenges where angles determine optimal designs.

Calculator Usage in Mathematics

Using a calculator can simplify many mathematical tasks, especially when dealing with inverse trigonometric functions. Today's calculators are equipped with functions like arcsine, arccosine, and arctangent accessible directly from a menu or keypad. Here's a quick guide on using them for finding \( \sec^{-1}(x) \):

• Transform the secant problem into a cosine one because most calculators readily support \( \cos^{-1} \).
• Input the reciprocal of your given secant value. For instance, if \( \sec^{-1}(-2.222) \), input \( \cos^{-1}(-0.450) \).
• Ensure your calculator is set to the desired units (radians or degrees) which align with your problem's context.

Understanding these functions deeper allows for quick conversions and greater accuracy in calculations. Remember, while calculators offer powerful tools, knowing the underlying mathematics helps interpret results and resolve intricacies like domain issues or possible extraneous solutions encountered in problems.