Problem 113 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-2, \sec \theta=\frac{1}{2}$$

Step-by-Step Solution

Answer

Both statements are impossible because the values are outside the valid range for cosine and secant.

Step 1: Understanding Cosine Range

The cosine of an angle, \( \cos \theta \), is defined to range between -1 and 1 inclusive. This means for any angle \( \theta \), the cosine value should satisfy \(-1 \leq \cos \theta \leq 1\).

Step 2: Evaluating \(\cos \theta = -2\)

Given \( \cos \theta = -2 \), observe that this value is outside the permissible range of cosine values. This implies that it is impossible for any angle \( \theta \) to have a cosine value of -2.

Step 3: Understanding Secant and its Relationship with Cosine

The secant function is defined as the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, for secant values, its corresponding cosine value \( \cos \theta \) must be within the standard range \(-1 \leq \cos \theta \leq 1\) and must not be zero to avoid division by zero.

Step 4: Evaluating \(\sec \theta = \frac{1}{2}\)

The equation \( \sec \theta = \frac{1}{2} \) implies \( \cos \theta = \frac{1}{\frac{1}{2}} = 2 \). A cosine value of 2 is outside the valid range of [-1, 1]. Thus, it is impossible for any angle \( \theta \) to satisfy \( \sec \theta = \frac{1}{2} \).

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, indicating the horizontal component of an angle in the unit circle.

• The cosine of an angle \( heta \) is typically defined using the adjacent side and hypotenuse of a right triangle as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• However, in a broader sense, its most essential range is between -1 and 1, and this applies to all angles \( \theta \) in the domain.

This restriction is due to its geometric representation on a unit circle, where the hypotenuse is always 1.An important point to keep in mind is:

• When the cosine value extends beyond this range, like \( \cos \theta = -2 \), it signifies an error or impossibility due to the mathematical and geometric fundamentals of the cosine function.

Such values indicate the need to re-examine the context or calculations leading to this result.

Secant Function

The secant function, labeled as \( \sec \theta \), is another crucial part of trigonometry. This function is defined as the reciprocal of the cosine function.

• Mathematically, it is expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
• This relationship implies that the domain of the secant function is inherently tied to the range of the cosine.

Due to being the reciprocal, certain rules apply:For secant values to be valid, the following must hold true:

• The cosine value it derives from cannot be zero, as division by zero is undefined.
• The implied cosine value from any secant calculation must lie between -1 and 1. For example, \( \sec \theta = \frac{1}{2} \) would mean \( \cos \theta = 2 \), which is not possible due to \( 2 \) being outside the valid range.

Thus, \( \sec \theta \) values must be verified through their cosine origins to ensure their mathematical legitimacy.

Angle Measurement

In trigonometry, angles can be expressed in several forms – degrees and radians being the most common.

• Degrees: A circle is divided into 360 degrees, a historically intuitive way to express angles.
• Radians: Based on the mathematical properties of circles, a full circle is \( 2\pi \) radians, making this measurement useful in calculus and other higher mathematics due to its natural relation to arc length.

Understanding these forms of measurement is crucial when working with trigonometric functions, as they often determine the forms outputs take.Additionally, regardless of the method of measuring angles, trigonometric functions such as cosine and secant will follow their inherent rules and limitations on these angles:

• These include the range and domain restrictions rooted in their geometric definitions.

Recognizing these will help in correctly interpreting and employing trigonometric results effectively.