Determine whether each statement is possible or impossible. c. cos θ = 5

Verified step by step guidance

1. Recall that the cosine function, \( \cos \theta \), represents the ratio of the adjacent side to the hypotenuse in a right triangle.
2. Understand that the range of the cosine function for real numbers is \([-1, 1]\).
3. Recognize that a value of \( \cos \theta = 5 \) is outside this range.
4. Conclude that it is impossible for \( \cos \theta \) to equal 5 for any real angle \( \theta \).
5. Therefore, the statement \( \cos \theta = 5 \) is impossible.

Key Concepts

Range of the Cosine Function

The cosine function, denoted as cos(θ), outputs values that range from -1 to 1 for any angle θ. This means that any statement claiming cos(θ) equals a value outside this range, such as 5, is impossible. Understanding this range is crucial for evaluating the validity of trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. Familiarity with these identities helps in simplifying and solving trigonometric equations. In this case, recognizing that cos(θ) cannot equal 5 is a direct application of the fundamental properties of trigonometric functions.

Understanding Angles and Their Functions

In trigonometry, angles can be measured in degrees or radians, and each angle corresponds to a specific value of sine, cosine, and tangent. Knowing how these functions behave at various angles aids in understanding their limits. Since cos(θ) is defined for all angles but constrained to the range of -1 to 1, it reinforces that certain values, like 5, are not achievable.