Problem 48 - Precalculus

Question

Determine whether each statement is possible or not. $$\cos \theta=1.0001$$

Step-by-Step Solution

Answer

The statement \(\cos \theta = 1.0001\) is impossible because cosine values are always between -1 and 1.

Step 1: Understanding the Cosine Function

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. In trigonometry, the cosine function, denoted as \(\cos \theta\), can take values between -1 and 1, inclusive, for all real numbers \(\theta\). It means \(\cos \theta \in [-1, 1]\).

Step 2: Analyzing the Given Statement

The statement suggests that \(\cos \theta = 1.0001\). We need to determine if this value falls within the range of valid cosine values, \([-1, 1]\).

Step 3: Comparing with the Valid Range

Since \(1.0001\) is greater than the maximum possible value \(1\) for \(\cos \theta\), this value exceeds the valid range \([-1, 1]\). Therefore, \(\cos \theta = 1.0001\) is not possible.

Step 4: Conclusion

Given that the cosine function cannot exceed a value of 1, the statement \(\cos \theta = 1.0001\) is impossible.

Key Concepts

Cosine Function

The cosine function is an essential trigonometric function that relates the angles and sides of a right triangle. This function is usually represented as \( \cos \theta \), where \( \theta \) is the angle. Cosine helps us understand how the angle relates to the triangle's side lengths:

• Adjacent side: The side next to the angle of interest.
• Hypotenuse: The longest side of the triangle, opposite the right angle.

The cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse. This makes the formula:\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] Cosine is vital for understanding angles and for calculations in various fields, including physics, engineering, and computer science. It's used not only in triangles but also in mappings like rotation in various dimensional spaces which involves knowing how angles are positioned efficiently.

Valid Range for Cosine

The cosine function has a specific range of values it can take. When dealing with real number angles, \( \cos \theta \) can vary from -1 to 1. This limitation is due to the definition of cosine as a ratio of sides in a right triangle. Let's look at why this is the case:

• When \( \theta = 0 \), the adjacent side equals the hypotenuse, making \( \cos \theta = 1 \).
• For \( \theta = 180^\circ \) or \( \theta = \pi \) radians, \( \cos \theta = -1 \).
• Values between these extremes will range between -1 and 1 as the angle increases from 0 through 180 degrees.

This range ensures the function remains consistent with its geometric interpretation, thus guaranteeing all cosine outcomes are logically sound. When we see values suggested outside this range, they are not possible in the framework of standard real-world trigonometry.

Right Triangle Properties

Right triangles have a remarkable set of properties that make them particularly important for trigonometry. These triangles contain one angle that is exactly 90 degrees, often marked by a small square box denoting the right angle. Key properties include:

• One angle in a right triangle is always 90 degrees, making the sum of all angles 180 degrees.
• The longest side in a right triangle, opposite the right angle, is the hypotenuse.
• The Pythagorean theorem applies, stating that the square of the hypotenuse length is equal to the sum of the squares of the other two sides: \( a^2 + b^2 = c^2 \).

These properties form the basis for understanding trigonometric ratios like sine, cosine, and tangent. They allow us to determine side lengths and angles, making right triangles essential in fields ranging from construction to navigation. By understanding these properties, we gain powerful tools for solving practical and theoretical problems.