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

Verified step by step guidance

1. Understand the range of the cosine function: The cosine of an angle \( \theta \) in a right triangle or on the unit circle is defined as the ratio of the adjacent side to the hypotenuse, or as the x-coordinate of a point on the unit circle.
2. Recall the range of the cosine function: The cosine function, \( \cos \theta \), has a range of \([-1, 1]\). This means that for any angle \( \theta \), the value of \( \cos \theta \) must be between -1 and 1, inclusive.
3. Evaluate the given statement: The statement \( \cos \theta = 1.5 \) suggests that the cosine value is 1.5, which is outside the range of possible values for the cosine function.
4. Determine the possibility: Since 1.5 is not within the range \([-1, 1]\), it is impossible for \( \cos \theta \) to equal 1.5.
5. Conclude: Therefore, the statement \( \cos \theta = 1.5 \) is impossible.

Key Concepts

Range of Cosine Function

The cosine function, denoted as cos(θ), outputs values strictly within the range of -1 to 1 for any angle θ. This means that any value outside this interval, such as 1.5, is not possible for the cosine of an angle. Understanding this range is crucial for determining the validity of statements involving cosine.

Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles. The primary functions include sine, cosine, and tangent, each defined based on the relationships between the angles and sides. Recognizing how these functions operate helps in evaluating statements about their possible values.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the values of sine and cosine for all angles. It is a circle with a radius of one centered at the origin of a coordinate plane. The x-coordinate of a point on the unit circle represents the cosine of the angle, reinforcing that cosine values must lie between -1 and 1.