Problem 125 - Thomas Calculus in SI Units

Question

Find the values of the following. a. \(\sec ^{-1} 1.5\) b. \(\csc ^{-1}(-1.5)\) c. \(\cot ^{-1} 2\)

Step-by-Step Solution

Answer

a. 0.8411 radians, b. -0.7297 radians, c. 0.4636 radians.

Step 1: Understanding the Problem - Part a

We need to find the inverse-secant of 1.5, denoted as \( \sec^{-1}(1.5) \). This requires us to find an angle \( \theta \) such that \( \sec(\theta) = 1.5 \), where \( 0 \leq \theta < \frac{\pi}{2} \) or \( \pi \leq \theta < \frac{3\pi}{2} \).

Step 2: Solving for \( \theta \) - Part a

Knowing \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we have \( \cos(\theta) = \frac{1}{1.5} \). Calculate \( \cos(\theta) \): \( \cos(\theta) = \frac{2}{3} \). So, \( \theta = \cos^{-1}\left(\frac{2}{3}\right) \).

Step 3: Understanding the Problem - Part b

For \( \csc^{-1}(-1.5) \), we need to find \( \theta \) such that \( \csc(\theta) = -1.5 \). The range for \( \csc^{-1}(x) \) is \(-\frac{\pi}{2} \leq \theta < 0 \) or \( 0 < \theta \leq \frac{\pi}{2} \).

Step 4: Solving for \( \theta \) - Part b

Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have \( \sin(\theta) = -\frac{1}{1.5} \). Calculate \( \sin(\theta) \): \( \sin(\theta) = -\frac{2}{3} \), so \( \theta = \sin^{-1}\left(-\frac{2}{3}\right) \).

Step 5: Understanding the Problem - Part c

We need to find \( \cot^{-1}(2) \), which requires finding an angle \( \theta \) such that \( \cot(\theta) = 2 \). The range for \( \cot^{-1}(x) \) is \( 0 < \theta < \pi \).

Step 6: Solving for \( \theta \) - Part c

Since \( \cot(\theta) = \frac{1}{\tan(\theta)} \), we have \( \tan(\theta) = \frac{1}{2} \). Therefore, \( \theta = \tan^{-1}\left(\frac{1}{2}\right) \).

Step 7: Final Answers

Evaluating each expression gives us approximate results: Part a: \( \sec^{-1}(1.5) \approx \cos^{-1}\left(\frac{2}{3}\right) \approx 0.8411 \) radians. Part b: \( \csc^{-1}(-1.5) \approx \sin^{-1}\left(-\frac{2}{3}\right) \approx -0.7297 \) radians. Part c: \( \cot^{-1}(2) \approx \tan^{-1}\left(\frac{1}{2}\right) \approx 0.4636 \) radians.

Key Concepts

Secant Function

Inverse trigonometric functions are used to find angles when given a trigonometric ratio. For the secant function, its inverse is denoted as \( \sec^{-1} \). The secant of an angle \( \theta \), denoted as \( \sec(\theta) \), is the reciprocal of the cosine, \( \frac{1}{\cos(\theta)} \). This means you find \( \theta \) such that \( \sec(\theta) = x \) by using the relation \( \cos(\theta) = \frac{1}{x} \).

Here are some important aspects to remember:

• The range for \( \sec^{-1}(x) \) is \( 0 \leq \theta < \frac{\pi}{2} \) or \( \pi \leq \theta < \frac{3\pi}{2} \).
• This range ensures that the cosine function is always non-zero, as \( \sec(\theta) \) becomes undefined if \( \cos(\theta) = 0 \).
• To solve \( \sec^{-1}(1.5) \), set it up as \( \cos(\theta) = \frac{2}{3} \) and find \( \theta \) using a calculator or a cosine table.

Cosecant Function

Inverse cosecant, \( \csc^{-1} \), is another inverse trigonometric function that helps us find angles. The cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function. So, \( \csc(\theta) = \frac{1}{\sin(\theta)} \). To find an angle when given \( \csc(\theta) = x \), we rearrange the relation to \( \sin(\theta) = \frac{1}{x} \).

Key Points for \( \csc^{-1} \):

• The range is \(-\frac{\pi}{2} \leq \theta < 0 \) or \( 0 < \theta \leq \frac{\pi}{2} \), ensuring sine is never zero.
• To find \( \csc^{-1}(-1.5) \), calculate \( \sin(\theta) = -\frac{2}{3} \) and determine \( \theta \) using \( \sin^{-1} \).
• This calculation means finding the angle whose sine is negative, fitting into the range where sine is expected to be negative.

Cotangent Function

The inverse cotangent function, denoted as \( \cot^{-1} \), is designed to find an angle given the cotangent value. Cotangent of an angle, \( \cot(\theta) \), is the reciprocal of the tangent function, defined as \( \frac{1}{\tan(\theta)} \). Solving for \( \theta \) requires reversing this relationship: if \( \cot(\theta) = x \), then \( \tan(\theta) = \frac{1}{x} \).

Important Considerations for \( \cot^{-1} \):

• The range for \( \cot^{-1}(x) \) is \( 0 < \theta < \pi \).
• This range ensures \( \tan(\theta) \) can take any real value, avoiding undefined behavior due to division by zero.
• For example, \( \cot^{-1}(2) \) requires finding \( \theta \) such that \( \tan(\theta) = \frac{1}{2} \), and this can be calculated using a tan inverse function.
• An inverse tangent calculation here shows which angle fits the \( \tan(\theta) \) requirement.