Final answer:
The trigonometric function of the quadrant angle secθ is undefined, thus the correct option is C.
Step-by-step explanation:
The trigonometric function of the quadrant angle secθ is undefined. This means that there is no numerical value that can be assigned to this expression. In other words, there is no real number that can be substituted for θ that would make the expression true. This is because the secant function is undefined at certain angles, specifically when θ is equal to 90° or 270°.
To understand why the secant function is undefined at these angles, we need to look at the definition of secant. The secant function is defined as the reciprocal of the cosine function, or secθ = 1/cosθ. In the first quadrant, where 0° < θ < 90°, the cosine function is positive and therefore the secant function is defined. However, in the second quadrant, where 90° < θ < 180°, the cosine function is negative and therefore the secant function is undefined. This same pattern continues in the third and fourth quadrants, where the cosine function is respectively negative and positive, resulting in the secant function being undefined.
Another way to understand why the secant function is undefined at 90° and 270° is by looking at the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin of the Cartesian plane. The x-coordinate of a point on the unit circle is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle. At 90° and 270°, the point on the unit circle has an x-coordinate of 0, meaning that the cosine of these angles is 0. Since the secant function is defined as 1/cosθ, and division by 0 is undefined, the secant function is also undefined at these angles.
In conclusion, the quadrant angle secθ is undefined because the secant function is undefined at 90° and 270°. This is due to the definition of the secant function and the behavior of the cosine function on the unit circle. Therefore, the correct answer to the given question is C) Undefined.