Problem 44 - Precalculus Mathematics for Calculus

Question

Most calculators do not have keys for \(\sec ^{-1}, \mathrm{csc}^{-1},\) or \(\mathrm{cot}^{-1}\) Prove the following identities, and then use these identities and a calculator to find \(\sec ^{-1} 2, \csc ^{-1} 3,\) and \(\cot ^{-1} 4\). $$\begin{array}{ll} \sec ^{-1} x=\cos ^{-1}\left(\frac{1}{x}\right), & x \geq 1 \\ \csc ^{-1} x=\sin ^{-1}\left(\frac{1}{x}\right), & x \geq 1 \\ \cot ^{-1} x=\tan ^{-1}\left(\frac{1}{x}\right), & x>0 \end{array}$$

Step-by-Step Solution

Answer

\(\sec^{-1}(2) \approx 1.047\) rad, \(\csc^{-1}(3) \approx 0.3398\) rad, \(\cot^{-1}(4) \approx 0.2449\) rad.

Step 1: Understand Reciprocal Trigonometric Functions

Recall that secant (sectheta), cosecant (csctheta) and cotangent (cottheta) are the reciprocal functions of cosine, sine, and tangent, respectively. This means \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\).

Step 2: Prove Identity for \(\sec^{-1}x\)

To find \(\sec^{-1} x\), we know from the reciprocal identity that \(\sec \theta = x\) implies \(\cos \theta = \frac{1}{x}\). Therefore, \(\theta = \cos^{-1}\left(\frac{1}{x}\right)\) when \(x \geq 1\), thus proving \(\sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right)\).

Step 3: Prove Identity for \(\csc^{-1}x\)

We want \(\csc^{-1} x\), and using the reciprocal identity \(\csc \theta = x\) implies \(\sin \theta = \frac{1}{x}\). Thus, \(\theta = \sin^{-1}\left(\frac{1}{x}\right)\) when \(x \geq 1\), proving \(\csc^{-1} x = \sin^{-1}\left(\frac{1}{x}\right)\).

Step 4: Prove Identity for \(\cot^{-1}x\)

To find \(\cot^{-1} x\), we recall the identity \(\cot \theta = x\) which implies \(\tan \theta = \frac{1}{x}\). Thus, \(\theta = \tan^{-1}\left(\frac{1}{x}\right)\) when \(x > 0\), proving \(\cot^{-1} x = \tan^{-1}\left(\frac{1}{x}\right)\).

Step 5: Use Calculator for \(\sec^{-1} 2\)

Using the identity \(\sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right)\), calculate \(\cos^{-1}\left(\frac{1}{2}\right)\), which gives \(\theta \approx 1.047\) radians or \(60^{\circ}\).

Step 6: Use Calculator for \(\csc^{-1} 3\)

Using the identity \(\csc^{-1} x = \sin^{-1}\left(\frac{1}{x}\right)\), calculate \(\sin^{-1}\left(\frac{1}{3}\right)\), resulting in \(\theta \approx 0.3398\) radians or \(19.47^{\circ}\).

Step 7: Use Calculator for \(\cot^{-1} 4\)

Using the identity \(\cot^{-1} x = \tan^{-1}\left(\frac{1}{x}\right)\), calculate \(\tan^{-1}\left(\frac{1}{4}\right)\), which gives \(\theta \approx 0.2449\) radians or \(14.04^{\circ}\).

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are special functions that you encounter when dealing with trigonometric ratios. They are called reciprocal because they involve flipping the values of sine, cosine, and tangent functions. The three main reciprocal trigonometric functions are: \( \sec \theta = \frac{1}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).
These functions are essential because not all calculators come with direct buttons for inverse reciprocal functions like \( \sec^{-1}, \csc^{-1} \), or \( \cot^{-1} \), which you might need in specific mathematical problems. Knowing the basics of these reciprocal identities can help you switch back to the basic trigonometric functions when calculating, making use of existing calculator functions. Understanding these reciprocal relationships provides a broader comprehension of trigonometry, making it easier to tackle exercises and problems in various mathematical fields.

Trigonometric Identities

Trigonometric identities are fundamental truths about trigonometric functions that hold for every value in their domains. They are like the rules or formulas that we can use to simplify and solve trigonometric equations. Here, inverse trigonometric identities become useful.
For instance:

• \( \sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right)\) for \( x \geq 1 \)
• \( \csc^{-1} x = \sin^{-1}\left(\frac{1}{x}\right)\) for \( x \geq 1 \)
• \( \cot^{-1} x = \tan^{-1}\left(\frac{1}{x}\right)\) for \( x > 0 \)

These identities are crucial when using standard trigonometric calculators that lack dedicated inverse keys for \( \sec, \csc, \) and \( \cot \). They allow us to swap the operation into a form that the calculator can handle, such as converting a secant inverse calculation into a cosine inverse calculation, which is typically supported by most calculators.

Using Calculators for Trigonometric Calculations

Calculators are highly useful tools for trigonometric calculations, especially when you need precise values for angles or trigonometric functions quickly. However, not every calculator will have buttons for every function you might need, such as \( \sec^{-1}(x), \csc^{-1}(x), \) and \( \cot^{-1}(x) \). In these instances, you can use calculator-friendly identities to help you find the solutions.
Consider the problem with \( \sec^{-1} 2 \). A typical calculator might not have a \( \sec^{-1} \) button, but it will almost certainly have \( \cos^{-1} \). Use the identity \( \sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right) \). So input \( \cos^{-1}\left(\frac{1}{2}\right) \) into your calculator to get \( \theta \approx 1.047 \) radians. Similarly, for \( \csc^{-1} 3 \), use \( \sin^{-1}\left(\frac{1}{3}\right) \) on your calculator to find \( \theta \approx 0.3398 \) radians, and for \( \cot^{-1} 4 \), try \( \tan^{-1}\left(\frac{1}{4}\right) \), resulting in \( \theta \approx 0.2449 \) radians. This approach ensures the efficient use of calculator functions you have, letting you overcome the limitations of the calculator and apply your understanding of inverse trigonometric identities.