Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , given that cos θ = 3/2 . "

Verified step by step guidance

1. Step 1: Recall the definition of secant in terms of cosine. The secant function is the reciprocal of the cosine function, so \( \sec \theta = \frac{1}{\cos \theta} \).
2. Step 2: Substitute the given value of \( \cos \theta = \frac{3}{2} \) into the reciprocal formula for secant.
3. Step 3: Recognize that the cosine of an angle cannot be greater than 1, as the range of the cosine function is \([-1, 1]\).
4. Step 4: Identify the error in the problem: \( \cos \theta = \frac{3}{2} \) is not possible for any real angle \( \theta \).
5. Step 5: Conclude that the problem is incorrect because it provides an impossible value for \( \cos \theta \).

Key Concepts

Definition of Secant

Secant is defined as the reciprocal of the cosine function. Mathematically, sec(θ) = 1/cos(θ). Understanding this relationship is crucial for solving problems involving secant, as it directly relates to the value of cosine.

Range of the Cosine Function

The cosine function has a range of values between -1 and 1, meaning that cos(θ) can never exceed 1 or be less than -1. Therefore, a value like 3/2 is invalid for cos(θ), indicating a fundamental misunderstanding of the function's properties.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Recognizing that sec(θ) cannot be calculated from an invalid cosine value requires familiarity with these identities and their constraints.