CONCEPT PREVIEW Determine whether each statement is possible or impossible. cos θ = 1.5

Verified step by step guidance

1. Recall the range of the cosine function: for any angle \( \theta \), \( \cos \theta \) must satisfy \( -1 \leq \cos \theta \leq 1 \).
2. Analyze the given value \( \cos \theta = 1.5 \) and compare it to the allowed range of cosine values.
3. Since \( 1.5 \) is greater than the maximum possible value of 1 for cosine, conclude that this value is outside the range of the cosine function.
4. Therefore, it is impossible for \( \cos \theta \) to equal 1.5 for any real angle \( \theta \).
5. This means the statement \( \cos \theta = 1.5 \) is impossible.

Key Concepts

Range of the Cosine Function

The cosine function outputs values only within the range of -1 to 1 for all real angles θ. Any value outside this interval, such as 1.5, is not possible for cos θ, indicating no real angle satisfies the equation.

Definition of Cosine in the Unit Circle

Cosine of an angle θ corresponds to the x-coordinate of the point on the unit circle at that angle. Since the unit circle has radius 1, the x-coordinate cannot exceed 1 or be less than -1, reinforcing the range limitation.

Implications of Impossible Trigonometric Values

When a trigonometric equation yields a value outside the function's range, it means no real solution exists. This helps in quickly determining the feasibility of equations like cos θ = 1.5 without further calculations.