Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-18, use a calculator to approximate each value. \(\sec ^{-1}(-2.222)\)

Step-by-Step Solution

Answer

\( \sec^{-1}(-2.222) \approx 116.57^\circ \) or \( 2.034 \) radians.

Step 1: Understand the Meaning of Secant Inverse

The function \( \sec^{-1}(x) \) represents the angle \( \theta \) whose secant is \( x \). So, we are looking for an angle \( \theta \) such that \( \sec(\theta) = -2.222 \).

Step 2: Recall Relationship with Cosine

We know that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, \( \cos(\theta) = \frac{1}{-2.222} \). Calculating this gives us \( \cos(\theta) \approx -0.450 \).

Step 3: Use Calculator to Find Angle

Using a calculator, take the inverse cosine of \( -0.450 \) to find \( \theta \). Make sure your calculator is in the correct mode (degrees or radians as required). The inverse cosine of \( -0.450 \) yields \( \theta \approx 116.57^\circ \) (if in degrees) or \( \theta \approx 2.034 \) radians.

Key Concepts

Arcsecant

Arcsecant, commonly written as \( \sec^{-1}(x) \), is the inverse function of the secant function. In trigonometry, when we talk about inverse functions, we are finding the angle that corresponds to a given trigonometric value. With arcsecant, this means finding an angle whose secant is \( x \).
Remember that secant is the reciprocal of cosine. So, if \( \sec(\theta) = x \), then \( \cos(\theta) = \frac{1}{x} \). This relationship helps us understand the process of finding arcsecant values.
If you're trying to find the arcsecant of a number like \(-2.222\), you are looking for an angle \( \theta \) such that \( \sec(\theta) = -2.222 \). This requires some calculation since it involves finding the reciprocal first and then using inverse trigonometric functions.

Trigonometric Functions

Trigonometric functions are a set of functions related to angles and triangles. They are fundamental in geometry, physics, engineering, and many other fields. The main trigonometric functions are sine, cosine, and tangent, and each has a corresponding reciprocal function: cosecant, secant, and cotangent, respectively.

• Sine (\( \sin \)) and cosecant (\( \csc \))
• Cosine (\( \cos \)) and secant (\( \sec \))
• Tangent (\( \tan \)) and cotangent (\( \cot \))

Understanding these functions and their relationships is crucial for dealing with inverses, such as arcsecant. For example, knowing that the secant of an angle is the reciprocal of the cosine allows us to calculate values and inverses.
These functions are periodic and repeat their values over regular intervals, which is important when determining angles associated with specific function values.

Inverse Functions

Inverse functions "reverse" the original function's process. For trigonometric functions, they allow us to determine angles based on their trigonometric values. The inverse trigonometric functions include arcsine (\( \sin^{-1} \)), arccosine (\( \cos^{-1} \)), and arctangent (\( \tan^{-1} \)).

• Arccosine, which helps find angles when you know the cosine value
• Arcsine, used to find angles from sine values
• Arctangent, used with tangent values

Similarly, the inverse secant, or arcsecant (\( \sec^{-1} \)), finds the angle when given the secant value. When you input a value like \(-2.222\), you're working backward to arrive at the corresponding angle. This process often involves a calculator, where the calculated cosine is converted to an angle using the inverse cosine function.
Inverse functions are only determined for specific ranges to ensure each function provides a unique result, which is essential for correctly identifying angles.