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. Recall the definition of the cosine function: \(\cos \theta\) represents the ratio of the adjacent side to the hypotenuse in a right triangle, and its value must lie between -1 and 1 inclusive.
2. Examine the given value: \(\cos \theta = \frac{3}{2} = 1.5\), which is greater than 1.
3. Since \(\cos \theta\) cannot be greater than 1 or less than -1 for any real angle \(\theta\), the given value is not possible for any real angle.
4. Therefore, the problem is flawed because it asks to find \(\sec \theta\) based on an invalid cosine value.
5. To correct the problem, ensure that the given value of \(\cos \theta\) is within the valid range \([-1, 1]\) before asking to find \(\sec \theta\).

Key Concepts

Domain of the Cosine Function

The cosine of an angle in trigonometry represents the ratio of the adjacent side to the hypotenuse in a right triangle, and its value must lie between -1 and 1. A value like 3/2 (which is 1.5) is outside this range, making it impossible for cos θ to equal 3/2 in real numbers.

Definition of Secant Function

The secant function, sec θ, is defined as the reciprocal of the cosine function, i.e., sec θ = 1/cos θ. To find sec θ, the cosine value must be valid and non-zero; otherwise, sec θ is undefined or does not exist in the real number system.

Validity of Trigonometric Values in Problems

When solving trigonometric problems, it is essential to verify that given values are within the permissible range for the function involved. Providing an invalid cosine value like 3/2 indicates a conceptual or typographical error, which must be identified before attempting to solve the problem.