If sec(−5π/4) = x, where the angle is measured in radians, find the exact value of x.

Introduction / Context:
This question assesses your understanding of trigonometric functions in radian measure and the even or odd nature of these functions. The secant function is the reciprocal of cosine, so evaluating sec(−5π/4) requires understanding cos(−5π/4) and the symmetry of cosine on the unit circle. Many exam questions use negative angles or angles outside the basic range to test whether you can use periodicity and symmetry correctly.

Given Data / Assumptions:
- The angle is −5π/4 radians, which is a negative angle.
- x is defined as sec(−5π/4).
- We must find the exact value of x without using decimal approximations.

Concept / Approach:
Cosine is an even function, meaning cos(−θ) = cos θ. Therefore, sec(−θ) = 1 / cos(−θ) = 1 / cos θ = sec θ. So we can first convert the negative angle into its positive counterpart and then evaluate using the unit circle. The angle 5π/4 corresponds to 225 degrees, which lies in the third quadrant where cosine is negative and has magnitude √2 / 2 for this special angle.

Step-by-Step Solution:
Step 1: Use the even nature of cosine: cos(−5π/4) = cos(5π/4). Step 2: Therefore sec(−5π/4) = 1 / cos(−5π/4) = 1 / cos(5π/4) = sec(5π/4). Step 3: Convert 5π/4 to degrees to reason more easily if needed: 5π/4 radians is 5 × 180 / 4 = 225 degrees. Step 4: The angle 225 degrees lies in the third quadrant, where both sine and cosine are negative. Step 5: The reference angle for 225 degrees is 45 degrees, and cos 45 degrees has magnitude √2 / 2. Step 6: Because we are in the third quadrant, cos 225 degrees = −√2 / 2. Step 7: Therefore cos(5π/4) = −√2 / 2, and sec(5π/4) = 1 / (−√2 / 2) = −2 / √2 = −√2. Step 8: Hence x = sec(−5π/4) = −√2.
Verification / Alternative check:
You can verify the sign and magnitude by recalling that the coordinates of the point on the unit circle at 225 degrees are (−√2 / 2, −√2 / 2). The x coordinate gives cosine directly, confirming cos(5π/4) = −√2 / 2. The reciprocal is sec(5π/4) = −√2. Since sec(−5π/4) is equal to sec(5π/4) due to even symmetry, this confirms the result.

Why Other Options Are Wrong:
Option a, −1/√3, corresponds to the reciprocal of a cosine value that would be √3, which does not match the special angle 225 degrees.
Option c, −1, would be correct for angles where cosine equals −1, such as π radians, but not for 5π/4.
Option d, √3, has the wrong sign and magnitude for this angle.
Option e, √2, has the correct magnitude but the wrong sign because cosine is negative in the third quadrant, making sec negative as well.

Common Pitfalls:
Students often forget which trigonometric functions are even and which are odd. Cosine and secant are even, while sine, tangent, and their reciprocals are odd. Another common mistake is misidentifying the correct quadrant for the angle, especially when it is given in radians. Carefully converting to degrees or using the unit circle can prevent these errors. Also, simplifying 1 / (−√2 / 2) must be done accurately to avoid losing the negative sign or mishandling the radical.

Final Answer:
The exact value of sec(−5π/4) is −√2.