Problem 59 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \sec^{-1}(2) = \cos^{-1}(0.5), \csc^{-1}(3) = \sin^{-1}(0.333), \cot^{-1}(4) = \tan^{-1}(0.25) \).

Step 1: Understanding the Problem

The exercise asks us to prove and use the identities linking inverse secant, cosecant, and cotangent functions to their respective cosine, sine, and tangent inverse functions. Our objective is to prove these identities and use a calculator to find values for \( \sec^{-1}(2), \csc^{-1}(3) \), and \( \cot^{-1}(4) \).

Step 2: Prove the Identity for sec Inverse

Recall \( \sec(A) = \frac{1}{\cos(A)} \). Therefore, \( \sec^{-1}(x) = A \) implies \( \cos(A) = \frac{1}{x} \). Hence, \( A = \cos^{-1}\left(\frac{1}{x}\right) \), which proves \( \sec^{-1} x = \cos^{-1}\left(\frac{1}{x}\right) \) for \( x \geq 1 \).

Step 3: Prove the Identity for csc Inverse

\( \csc(A) = \frac{1}{\sin(A)} \) implies \( \csc^{-1}(x) = A \) means that \( \sin(A) = \frac{1}{x} \). Hence, \( A = \sin^{-1}\left(\frac{1}{x}\right) \) for \( x \geq 1 \), proving \( \csc^{-1} x = \sin^{-1}\left(\frac{1}{x}\right) \).

Step 4: Prove the Identity for cot Inverse

\( \cot(A) = \frac{1}{\tan(A)} \) implies \( \cot^{-1}(x) = A \) means that \( \tan(A) = \frac{1}{x} \). Therefore, \( A = \tan^{-1}\left(\frac{1}{x}\right) \), proving \( \cot^{-1} x = \tan^{-1}\left(\frac{1}{x}\right) \) for \( x > 0 \).

Step 5: Use the Identity to Calculate sec Inverse

For \( \sec^{-1}(2) \), use \( \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \). Calculate \( \cos^{-1}\left(\frac{1}{2}\right) \) using a calculator, which yields approximately \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

Step 6: Use the Identity to Calculate csc Inverse

For \( \csc^{-1}(3) \), use \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \). With a calculator, compute \( \sin^{-1}\left(\frac{1}{3}\right) \), yielding approximately \( 19.47^\circ \) or \( 0.34 \) radians.

Step 7: Use the Identity to Calculate cot Inverse

For \( \cot^{-1}(4) \), use \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \). Using a calculator, find \( \tan^{-1}\left(\frac{1}{4}\right) \), resulting in approximately \( 14.04^\circ \) or \( 0.25 \) radians.

Key Concepts

secant_inverse

When tackling the computation of the secant inverse, it's essential to understand the relationship between secant and cosine functions. The key identity here is: \( \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \). This translates to finding the angle whose secant is \( x \) by evaluating the cosine of the reciprocal of \( x \). For direct calculations, this identity simplifies the process since most calculators lack a specific function for the secant inverse.

Here's how it works:

• Recognize the secant function as \( \sec(A) = \frac{1}{\cos(A)} \).
• The inverse secant function \( \sec^{-1}(x) \) identifies the angle \( A \) such that \( \cos(A) = \frac{1}{x} \).
• Thus, solving \( \sec^{-1}(2) \) involves calculating \( \cos^{-1}\left(\frac{1}{2}\right) \), which is approximately \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

This approach effectively leverages existing calculator functionality to compute values for the secant inverse by utilizing the cosine inverse.

cosecant_inverse

The inverse function of cosecant can be approached similarly to secant, using its relationship with sine. The central identity for calculating cosecant inverse is \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \). This means to find the angle whose cosecant equals \( x \), we look for the sine of the reciprocal of \( x \).

Here's the logic breakdown:

• The cosecant function is defined as \( \csc(A) = \frac{1}{\sin(A)} \).
• In inverse terms, \( \csc^{-1}(x) \) determines the angle \( A \) when \( \sin(A) = \frac{1}{x} \).
• To calculate \( \csc^{-1}(3) \), evaluate \( \sin^{-1}\left(\frac{1}{3}\right) \) which results in around \( 19.47^\circ \) or \( 0.34 \) radians.

Utilizing the sine inverse not only simplifies calculations but also helps in circumventing the limitation posed by the absence of a direct calculator key for cosecant inverse.

cotangent_inverse

Calculating the inverse cotangent involves understanding its link to the tangent function. The identity \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \) is foundational for these calculations. Essentially, to determine the angle whose cotangent is \( x \), we compute the tangent of the reciprocal of \( x \).

Step-by-step logic:

• Define the cotangent function as \( \cot(A) = \frac{1}{\tan(A)} \).
• The inverse cotangent, \( \cot^{-1}(x) \), finds \( A \) where \( \tan(A) = \frac{1}{x} \).
• For \( \cot^{-1}(4) \), compute \( \tan^{-1}\left(\frac{1}{4}\right) \), yielding approximately \( 14.04^\circ \) or \( 0.25 \) radians.

Employing this identity allows for utilizing the more universally available tangent inverse function, facilitating easier calculations of cotangent inverse when such direct keys are absent.