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The secant function, denoted as \( \sec x \), plays a crucial role in trigonometry. It is defined as the reciprocal of the cosine function, specifically \( \sec x = \frac{1}{\cos x} \). Understanding this definition is key, as it helps you see why the secant function is undefined wherever the cosine function equals zero. However, in the interval \( 0 < x < \frac{\pi}{2} \), the cosine function remains positive and non-zero, ensuring \( \sec x \) is well-defined.

When exploring the behavior of \( \sec x \), it's important to recognize its tendency to increase without bound as the angle \( x \) approaches points where \( \cos x \) approaches zero. This can be visualized clearly as \( x \) nears \( \frac{\pi}{2} \), causing \( \cos x \) to get very close to zero, and thus \( \sec x \) shoots up to positive infinity.

• The secant function is periodic with a period of \( 2\pi \).
• It is positive when the cosine is positive, aligning with intervals where \( \cos x > 0 \).
• The graph of \( \sec x \) exhibits vertical asymptotes at points where \( \cos x = 0 \), excluding these in the specified interval.

These characteristics make \( \sec x \) a fascinating function with unique properties beneficial in graph analysis.

The tangent function, represented by \( \tan x \), is another pivotal concept in trigonometry, given by the ratio of the sine function to the cosine function: \( \tan x = \frac{\sin x}{\cos x} \). This definition inherently means that the tangent function is undefined wherever the cosine function becomes zero. Fortunately, in the desired interval \( 0 < x < \frac{\pi}{2} \), the cosine function is positive, making \( \tan x \) valid throughout.

As \( x \) approaches \( \frac{\pi}{2} \), \( \cos x \) gets progressively smaller, pushing \( \tan x \) towards infinity. At the lower end of the interval, where \( x \to 0^{+} \), the tangent starts at zero because \( \sin x \approx 0 \) and \( \cos x \approx 1 \).

• The tangent function repeats every \( \pi \) interval, manifesting periodic characteristics over its domain.
• It is positive where both \( \sin x \) and \( \cos x \) share the same sign, notably in the first quadrant \( 0 < x < \frac{\pi}{2} \).
• Similarly to the secant function, \( \tan x \) will also show vertical asymptotes where \( \cos x = 0 \).

The increasing trend of \( \tan x \) as \( x \) approaches \( \frac{\pi}{2} \) is crucial for sketching its curve accurately.

Vertical asymptotes are lines that a curve approaches but never touches or crosses. They often appear in trigonometric functions where there is a division by zero in the function's expression, causing the values to increase or decrease indefinitely. For expressions involving \( \sec x \) and \( \tan x \), vertical asymptotes occur at points where the cosine function, \( \cos x \), equals zero.

In our context of \( 0 < x < \frac{\pi}{2} \), the significant vertical asymptote to consider is at \( x = \frac{\pi}{2} \). As \( x \) closes in on \( \frac{\pi}{2} \), both \( \sec x \) and \( \tan x \) approach infinity due to the denominator \( \cos x \) reaching closer to zero.

• Vertical asymptotes are indicative of points where a function heads towards infinity either positively or negatively.
• While the curve is never able to reach or cross the asymptote, it provides a boundary that helps in shaping the curve's path.
• In a sketch, vertical asymptotes are represented using dashed lines, signifying their role as imaginary barriers.

Understanding vertical asymptotes is crucial for predicting and visualizing the behavior of trigonometric function curves accurately.